Quantum mechanical framework for interaction of OAM with matter and applications in solid states, biosciences and quantum computing

ABSTRACT

A system for applying orbital angular momentum (OAM) to electrons of a semiconductor material comprises a light source generator for generating a plane wave light beam. Orbital angular momentum (OAM) processing circuitry applies at least one orbital angular momentum to the plan wave light beam to generate an OAM light beam. The OAM processing circuitry controls transitions of electrons between quantized states within the semiconductor material to perform quantum entanglement within the semiconductor material responsive to the at least one orbital angular momentum applied to the plane wave light beam. A transmitter transmits the OAM light beam at the semiconductor material to induce the transitions of the electrons between the quantize states and perform the quantum entanglement within the semiconductor material.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.16/660,246, entitled QUANTUM MECHANICAL FRAMEWORK FOR INTERACTION OF OAMWITH MATTER AND APPLICATIONS IN SOLID STATES, BIOSCIENCES AND QUANTUMCOMPUTING, filed on Oct. 22, 2019, which claims priority to U.S.Provisional Application No. 62/902,146, entitled OAM BEAM INTERACTIONSWITH MATTER FOR APPLICATIONS TO SOLID STATES, BIOSCIENCES AND QUANTUMCOMPUTING, filed on Sep. 18, 2019, the specifications of which areincorporated by reference herein in their entirety.

U.S. patent application Ser. No. 16/660,246 is a continuation-in-part ofU.S. patent application Ser. No. 16/509,301, entitled UNIVERSAL QUANTUMCOMPUTER, COMMUNICATION, QKD SECURITY AND QUANTUM NETWORKS USING OAMQU-DITS WITH DLP, filed on Jul. 11, 2019 and is a continuation-in-partof U.S. patent application Ser. No. 14/882,085, entitled APPLICATION OFORBITAL ANGULAR MOMENTUM TO FIBER, FSO AND RF, filed on Oct. 13, 2015,and is a continuation-in-part of U.S. patent application Ser. No.14/816,781, entitled SUPPRESSION OF ELECTRON-HOLE RECOMBINATION USINGORBITAL ANGULAR MOMENTUM SEMICONDUCTOR DEVICES, filed on Aug. 3, 2015,the specifications of which are incorporated by reference herein intheir entirety.

TECHNICAL FIELD

The present invention relates to controlling a movement of particleswithin a material, and more particularly to controlling the movement ofparticles using orbital angular momentum applied to photons within alight beam.

BACKGROUND

Within photonic devices, light energy is used for exciting electrons tohigher energy levels and creating electrical energy or currentsresponsive to the light energy. The ability to control the states ofelectrons in materials such as semiconductors, biological material andquantum computers provide a number of advantages with respect to thesematerials. Currently, available techniques suffer from a number ofdeficiencies. Thus, some manner for increasing the increasing theability to control electron states within various types of materialswould provide for a number of improvements in the use and operation ofthese materials.

SUMMARY

The present invention, as disclosed and described herein in one aspectthereof comprises a system for applying orbital angular momentum (OAM)to electrons of a semiconductor material comprises a light sourcegenerator for generating a plane wave light beam. Orbital angularmomentum (OAM) processing circuitry applies at least one orbital angularmomentum to the plan wave light beam to generate an OAM light beam. TheOAM processing circuitry controls transitions of electrons betweenquantized states within the semiconductor material to perform quantumentanglement within the semiconductor material responsive to the atleast one orbital angular momentum applied to the plane wave light beam.A transmitter transmits the OAM light beam at the semiconductor materialto induce the transitions of the electrons between the quantize statesand perform the quantum entanglement within the semiconductor material.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to thefollowing description taken in conjunction with the accompanyingDrawings in which:

FIG. 1A illustrates an overall view of various components of a universalquantum computer system using OAM qudits with DLP;

FIG. 1B illustrates a block diagram of a universal quantum computersystem using OAM qudits with DLP;

FIG. 2 illustrates the use of spin polarization for quantum gate inputs;

FIG. 3 illustrates the use of OAM for quantum gate inputs;

FIG. 4 illustrates various types of quantum logic circuits;

FIG. 5 illustrates a single bit rotation gate;

FIG. 6 illustrates a two bit controlled NOT gate; and

FIG. 7 illustrates a Qudit network.

FIG. 8 illustrates various techniques for increasing spectral efficiencywithin a transmitted signal;

FIG. 9 illustrates a particular technique for increasing spectralefficiency within a transmitted signal;

FIG. 10 illustrates a general overview of the manner for providingcommunication bandwidth between various communication protocolinterfaces;

FIG. 11 is a functional block diagram of a system for generating orbitalangular momentum within a communication system;

FIG. 12 is a functional block diagram of the orbital angular momentumsignal processing block of FIG. 6;

FIG. 13 is a functional block diagram illustrating the manner forremoving orbital angular momentum from a received signal including aplurality of data streams;

FIG. 14 illustrates a single wavelength having two quanti-spinpolarizations providing an infinite number of signals having variousorbital angular momentums associated therewith;

FIG. 15A illustrates an object with a spin angular momentum;

FIG. 15B illustrates an object with an orbital angular momentum;

FIG. 15C illustrates a circularly polarized beam carrying spin angularmomentum;

FIG. 15D illustrates the phase structure of a light beam carrying anorbital angular momentum;

FIG. 16A illustrates a plane wave having only variations in the spinangular momentum;

FIG. 16B illustrates a signal having both spin and orbital angularmomentum applied thereto;

FIGS. 17A-17C illustrate various signals having different orbitalangular momentum applied thereto;

FIG. 17D illustrates a propagation of Poynting vectors for various Eigenmodes;

FIG. 17E illustrates a spiral phase plate;

FIG. 18 illustrates a system for using the orthogonality of an HG modalgroup for free space spatial multiplexing;

FIG. 19 illustrates various manners for converting a Gaussian beam intoan OAM beam;

FIG. 20 illustrates a manner for generating a light beam includingorthogonal functions;

FIGS. 21A-21H illustrate holograms that may be used for modulating alight beam;

FIG. 22A is a block diagram of a digital micro-mirror device;

FIG. 22B illustrates the manner in which a micro-mirror interacts with alight source;

FIG. 23 illustrates the mechanical structure of the micro-mirror;

FIG. 24 is a block diagram of the functional components of amicro-mirror;

FIG. 25 illustrates a flow chart of the process for changing theposition of a micro-mirror;

FIG. 26 illustrates an intensity in phase interferometer for measuringthe intensity and phase of a generated beam;

FIG. 27A illustrates the manner in which switching between different OAMmodes may be achieved in real time;

FIG. 27B illustrates the manner in which a transmitter processesmultiple data channels that are passed through a cylindrical lens to afocusing lens;

FIG. 28 illustrates the window transmission curves for Corning 7056;

FIGS. 29-33 are zoomed in views of visible and UV AR coated windowtransmittance for Corning 7056;

FIG. 34 illustrates circuitry for the generation of an OAM twisted beamusing a hologram within a micro-electromechanical device:

FIG. 35 illustrates the use of multiple single holograms formultiplexing;

FIG. 36 illustrates various reduced binary for holograms for applyingOAM levels;

FIG. 37A illustrates the combined use of OAM and polarizationprocessing;

FIG. 37B illustrates basic quantum modules;

FIG. 38 illustrates examples of quantum gates;

FIG. 39 illustrates a qudit gate implemented using OAM degrees offreedom;

FIG. 40 illustrates an OAM based qudit teleportation module;

FIG. 41 illustrates a syndrome calculator module;

FIG. 42 illustrates a syndrome for identifying quantum error based onsyndrome measurements;

FIG. 43 illustrates a block diagram of a quantum computer;

FIG. 44 illustrates an E/O modulator;

FIG. 45 illustrates a generalized-CNOT gate;

FIG. 46 illustrates the operation of a CNOT gate on a quantum registerconsisting of two qubits;

FIG. 47 illustrates a block diagram of an OAM processing systemutilizing quantum key distribution;

FIG. 48 illustrates a basic quantum key distribution system;

FIG. 49 illustrates the manner in which two separate states are combinedinto a single conjugate pair within quantum key distribution;

FIG. 50 illustrates one manner in which 0 and 1 bits may be transmittedusing different basis within a quantum key distribution system;

FIG. 51 is a flow diagram illustrating the process for a transmittertransmitting a quantum key;

FIG. 52 illustrates the manner in which the receiver may receive anddetermine a shared quantum key;

FIG. 53 more particularly illustrates the manner in which a transmitterand receiver may determine a shared quantum key;

FIG. 54 is a flow diagram illustrating the process for determiningwhether to keep or abort a determined key;

FIG. 55 illustrates a functional block diagram of a transmitter andreceiver utilizing a free-space quantum key distribution system;

FIG. 56 illustrates a network cloud-based quantum key distributionsystem;

FIG. 57 illustrates a high-speed single photon detector in communicationwith a plurality of users;

FIG. 58 illustrates a nodal quantum key distribution network;

FIG. 59 illustrates a light beam;

FIG. 60 illustrates a plane wave front;

FIG. 61 illustrates a helical wave front;

FIG. 62 illustrates the use of light harvesting complexes (LHCs) forgenerating electron-hole pairs;

FIG. 63 illustrates the use of organic photovoltaic cells (OPVs) forgenerating electron-hole pairs;

FIG. 64 illustrates the effect of spin and decentralization onelectron-hole recombination;

FIG. 65 illustrates the effect of an OAM twisted beam on electron-holerecombination;

FIG. 66 illustrates a functional block diagram of one manner forimproving the suppression of electron-hole recombination in a photoniccircuit;

FIG. 67 illustrates the manner in which electrons move between differentstates within an organic photovoltaic cell;

FIG. 68 is a flow diagram illustrating the process for controllingelectron-hole recombination using the application of orbital angularmomentum to a light beam;

FIG. 69 illustrates a flow diagram of the process for using a light beamto apply OAM to electrons;

FIG. 70 illustrates a block diagram of a system for performing theoperations illustrated in FIG. 69;

FIG. 71 illustrates the manner in which various bands of a semiconductormay be excited by optical excitation;

FIG. 72 illustrates the use of orbital angular momentum in photons toapply orbital angular momentum to electrons creating electric currentsand magnetic field;

FIG. 73 illustrates a flow diagram of a process for determining totalcurrent/magnetic field provided by a group of electrons;

FIG. 74 illustrates a flow diagram for determining mean field fortwisted light and the effects of collision terms;

FIG. 75 illustrates a flow diagram for determining light beaminteractions for a bulk system using a solid imagined as a cylinder;

FIG. 76 illustrates electrons within a lattice of ions within a solid;

FIG. 77 illustrates a manner for encoding information with in a twist ofthe state function (wave function);

FIG. 78 illustrates an example of braiding anyons;

FIG. 79 illustrates a braid of anyon stands;

FIG. 80 illustrates a further example of a braid:

FIG. 81 illustrates interaction between OAM light and graphene;

FIG. 82 illustrates a graphene lattice in a Honeycomb structure; and

FIG. 83 illustrates a manner for using an OAM light beam four revisinggene sequences using CRISPR-CAS9 technologies.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are usedherein to designate like elements throughout, the various views andembodiments of a quantum mechanical framework for interaction of OAMwith applications in solid states, biosciences and quantum computing areillustrated and described, and other possible embodiments are described.The figures are not necessarily drawn to scale, and in some instancesthe drawings have been exaggerated and/or simplified in places forillustrative purposes only. One of ordinary skill in the art willappreciate the many possible applications and variations based on thefollowing examples of possible embodiments.

Referring now to FIG. 1A, there is illustrated an overall view of thevarious components contributing to a universal quantum computer systemusing OAM qudits with DLP. The present invention uses an approach forboth quantum communication and quantum computing applications 100 towork simultaneously using OAM Qudits for implementing Qu-dit Gates 102and Qu-dit Quantum Key Distribution (QKD) 104 with integrated photonics(i.e. DLP) 106. The OAM signals are generated using DLP 106 and allowedto utilize OAM propagation to propagate through optical fibers 108. TheOAM propagated signals are utilized within qu-dit gates 102 and may makeuse of qudit QKD 104 in order to perform quantum computing,communications and encryption 100. A qudit comprises a quantum unit ofinformation that may take any of d states, where d is a variable.

Referring now to FIG. 1B, there is illustrated a block diagram of auniversal quantum computer system using OAM qudits with DLP. An inputdata stream 120 is provided to OAM 122 to have OAM values applied to thedata 120. The OAM processing enables photons to carry an arbitrarynumber of bits per photon. The OAM processed data bits are applied tothe photons using DLP technologies using DLP processing 123 as describedhereinbelow. The signals from the DLP processing 123 are provided toqudit gates 124. The qudit gates 124 may comprise generalized X-gates,generalized Z-gates and generalized CNOT-gates that are qudit versionsof existing qubit gates. The qudit gates 124 may also comprise modulessuch as fault-tolerate quantum computing modules, QKD modules, etc. Themodules may provide for quantum error correction (i.e. non-binarysyndrome module); entanglement-assisted QKD (i.e. the generalizedBell-states, etc.). The basic qudit gate 124 would comprise a QFT(quantum Fourier transform). Thus, the F-gate on qudits has the sameeffect as the Hadamard gate on qubits |0> is mapped to 1/sqrt(2){|0>+|1>}, |1> is mapped to 1/sqrt(2) {|0>−|1>}. The signals out footfrom the cutie Gates 124 may then be used in for example a quantum keydistribution (QKD) process 126. Existing QKD with high-speedcommunications and computing is very slow. By using the system describedherein there may be a simultaneous increase insecurity and throughputwhile further increasing the capacity of computing and processing of thesystem.

The photon angular momentum of photons can be used to carry both thespin angular momentum (SAM) and the orbital angular momentum (OAM) totransmit multiple data bits. SAM is associated with polarization, whileOAM is associated with azimuthal phase dependence of the complexelectric field. Given that OAM eigenstates are mutually orthogonal, alarge number of bits per single photon can be transmitted. This is moreparticularly illustrated with respect to FIGS. 2 and 3. FIG. 2illustrates how polarization spin 202 may be applied to data bit values204 to generate qubit values 206. Since each data bit value 204 may onlyhave a positive spin polarization or a negative spin polarizationapplied thereto only a pair of qubit states are available for each datavalue. However, as shown in FIG. 3, if an OAM value 302 is applied toeach data bit value 304 a much larger number of qudit values 306 maybeobtain for each data bit value 304. The number of qudit values may rangefrom 1 to N, where N is the largest number of different OAM states thatare applied to the data values 304. The manner for applying the OAMvalues 302 to the data bit value 304 may use the DLP processing 123 thatwill be more fully described herein below.

Referring now back to FIG. 1, the OAM processed signals may betransmitted using OAM propagation in optical fibers 108. The ability togenerate/analyze states with different photon angular momentum appliedthereto, by using holographic methods, allows the realization of quantumstates in multidimensional Hilbert space. Because OAM states provide aninfinite basis state, while SAM states are 2-D only, the OAM states canalso be used to increase the security for quantum key distribution (QKD)applications 104 and improve computational power for quantum computingapplications 100. The goal of the system is to build angular momentumbased deterministic universal quantum qudit gates 102, namely,generalized-X, generalized-Z, and generalized-CNOT qudit gates, anddifferent quantum modules of importance for various applications,including fault-tolerant quantum computing, teleportation, QKD, andquantum error correction. For example, the basic quantum modules forquantum teleportation applications include the generalized-Bell-stategeneration module and the QFT-module. The basic module for entanglementassisted QKD is either the generalized-Bell-state generation module orthe Weyl-operator-module. The approach is to implement all these modulesin integrated optics using multi-dimensional qudits on DLP.

In quantum computing a qubit or quantum bit is the basic unit of quantuminformation and comprises the quantum version of the classical binarybit physically realized with a two-state device. A qubit is a two-statequantum-mechanical system, one of the simplest quantum systemsdisplaying the characteristics of quantum mechanics. Examples include:the spin of the electron in which the two levels can be taken as spin upand spin down; or the polarization of a single photon in which the twostates can be taken to be the vertical polarization and the horizontalpolarization. In a classical system, a bit would have to be in one stateor the other. However, quantum mechanics allows the qubit to be in acoherent superposition of both states/levels at the same time, aproperty that is fundamental to quantum mechanics and thus quantumcomputing.

In quantum computing, the concept of ‘qubit’ has been introduced as thecounterpart of the classical concept of ‘bit’ in conventional computers.The two qubit states labeled as |0> and |1> correspond to the classicalbits 0 and 1 respectively. The arbitrary qubit state |φ> maintains acoherent superposition of states |0> and |1>:|φ>=a|0>+b|1>where a and b are complex numbers called probability amplitudes. Thatis, the qubit state |φ> collapses into either |0> state with probability|a|², or |1> state with probability |b|² with satisfying |a|²+|b|²=1.

The qubit state |φ> is described as|φ>=cos θ>+e ^(iψ) sin θ|1>which is called Bloch-sphere representation. In order to give a minimumrepresentation of the basic quantum logic gates, this is rewritten as afunction with complex-valued representation, by corresponding theprobability amplitudes a and b as the real part and imaginary part ofthe function respectively. The quantum state with complex-valuedrepresentation is described asf(θ)=e ^(iθ)=cos θ+i sin θ,

In quantum computing and specifically the quantum circuit model ofcomputation, a quantum logic gate (or simply quantum gate) is a basicquantum circuit operating on a small number of qubits. They are thebuilding blocks of quantum circuits, like classical logic gates are forconventional digital circuits.

Quantum Gates

In quantum computing and specifically the quantum circuit model ofcomputation, a quantum logic gate (or simply quantum gate) is a basicquantum circuit operating on a small number of qubits. They are thebuilding blocks of quantum circuits, like classical logic gates are forconventional digital circuits.

Unlike many classical logic gates, quantum logic gates are reversible.However, it is possible to perform classical computing using onlyreversible gates. For example, the reversible Toffoli gate can implementall Boolean functions, often at the cost of having to use ancillarybits. The Toffoli gate has a direct quantum equivalent, showing thatquantum circuits can perform all operations performed by classicalcircuits.

Quantum logic gates are represented by unitary matrices. The most commonquantum gates operate on spaces of one or two qubits, just like thecommon classical logic gates operate on one or two bits. As matrices,quantum gates can be described by 2^(n)×2^(n) sized unitary matrices,where n is the number of qubits that the gates act on. The variablesthat the gates act upon, the quantum states, are vectors in 2^(n)complex dimensions, where n again is the number of qubits of thevariable. The base vectors are the possible outcomes if measured, and aquantum state is a linear combination of these outcomes.

Referring now to FIG. 4, in quantum logic circuits, fundamental quantumgates 3802 are the single bit rotation gate 404 and two-bit controlledNOT gate 406. Any quantum logic circuit can be constructed bycombinations of these two gates. As shown in FIG. 5, a single bitrotation gate 404 takes a quantum state as its input 502 and outputs arotated state in the complex plane at its output 504. This gate can bedescribed as f(θ₁+θ₂)=f(θ₁)·f(θ₂).

A two-bit controlled NOT gate 602, as shown in FIG. 6, takes two quantumstates as its inputs 602 and gives two outputs: one of the input states604 and the exclusive OR-ed result of two inputs 606. It is necessary torepresent the inversion and non-inversion of the quantum state in orderto describe this operation, thus a controlled input parameter V isintroduced:

${f\left( {{\frac{\pi}{2}\gamma} + {\left( {1 - {2\;\gamma}} \right) \cdot \theta}} \right)} = \left\{ \begin{matrix}{{{\cos\;\theta} + {i\;\sin\;\theta}},} & {\gamma = 0} \\{{{\sin\;\theta} + {i\;\cos\;\theta}},} & {\gamma = 1}\end{matrix} \right.$

The output state of the neuron k, denoted as x_(k), is given as:x _(k) =f(y _(k))=cos y _(k) +i sin y _(k) =e ^(iy) ^(k)A similar formulation for Qudits and a corresponding neural networkapproach may also be provided.

In recent years, scientists have developed quantum-neuro computing inwhich the algorithm of quantum computation is used to improve theefficiency of neural computing systems. The quantum state and theoperator of quantum computation are both important to realizeparallelisms and plasticity respectively in information processingsystems. The complex valued representation of these quantum conceptsallows neural computation system to advance in learning abilities and toenlarge its possibility of practical applications. The application ofthe above described application of nonlinear modeling and forecasting toAI may be implemented according to two specific proposals. One is to usea Qubit neural network model based on 2-dimensional Qubits and thesecond is to expand the Qubits to multi-dimensional Qudits and furtherinvestigate their characteristic features, such as the effects ofquantum superposition and probabilistic interpretation in the way ofapplying quantum computing to a neural network.

One of the applications for quantum networks is to predict time-seriesfrom dynamical systems, especially from chaotic systems by their variousapplications. There have been several attempts to use real-valued neuralnetworks for predictions, but there have been no attempts for predictionby complex-valued Qudit-based neural networks using nonlinear attractorreconstruction where learning iterations, learning success rates, andprediction errors may be examined. Thus, as shown generally in FIG. 7, aQudit based network 702 may implement nonlinear modeling and forecastingto AI 704 that generates a nonlinear attractor reconstructions 706 fromthe time-series data 708.

This process implements developments in quantum-neuro computing in whichthe algorithm of quantum computation is used to improve the efficiencyof a neural computing system and those can be used in conjunction withthe use of attractors to predict future behavior. The attractor approachis totally classical and not quantum mechanical. For example whendelaying embedding to reconstruct the attractor, it is one simpleprocess of delay embedding that occurs multiple times in parallel andtherefore quantum computation can be used to realize parallelisms inreal time to perform the process of delay embedding. The firstimplementation is to use Qubit neural network model based on2-dimensional Qubits to construct attractors and provide predictions offuture behavior as described herein and a second is to expand the Qubitsto multi-dimensional Qudits for the same purposes and furtherinvestigate their characteristic features, such as the effects ofquantum superposition and probabilistic interpretation in the way ofapplying quantum computing to neural network.

OAM Generation

Application of Orbital Angular Momentum to photons that are provided asinput to Quantum Gates enable greater amounts of data to each individualphoton. The use of OAM enables an arbitrary number of bits to be carriedper photon. Achieving higher data carrying capacity is perhaps one ofthe primary interest of the computing community. This is led to theinvestigation of using different physical properties of a light wave forcommunications and data transmission, including amplitude, phase,wavelength and polarization. Orthogonal modes in spatial positions arealso under investigation and seemed to be useful as well. Generallythese investigative efforts can be summarized in 2 categories: 1)encoding and decoding more bits on a single optical pulse; a typicalexample is the use of advanced modulation formats, which encodeinformation on amplitude, phase and polarization states, and 2)multiplexing and demultiplexing technologies that allow parallelpropagation of multiple independent data channels, each of which isaddressed by different light property (e.g., wavelength, polarizationand space, corresponding to wavelength-division multiplexing (WDM),polarization-division multiplexing (PDM) and space division multiplexing(SDM), respectively). One manner for achieving the higher data capacityis through using OAM communications and computing which is a process ofapplying orbital angular momentum to communications/quantum computingsignals to prevent interference between signals and to provide for anincreased bandwidth as described in U.S. patent application Ser. No.14/864,511, entitled APPLICATION OF ORBITAL ANGULAR MOMENTUM TO FIBER,FSO AND RF, which is incorporated herein by reference in its entirety.

The recognition that orbital angular momentum (OAM) has applications incommunication and quantum computing has made it an interesting researchtopic. It is well-known that a photon can carry both spin angularmomentum and orbital angular momentum. Contrary to spin angular momentum(e.g., circularly polarized light), which is identified by theelectrical field direction, OAM is usually carried by a light beam witha helical phase front. Due to the helical phase structure, an OAMcarrying beam usually has an annular intensity profile with a phasesingularity at the beam center. Importantly, depending on discretetwisting speed of the helical phase, OAM beams can be quantified isdifferent states, which are completely distinguishable while propagatingcoaxially. This property allows OAM beams to be potentially useful ineither of the two aforementioned categories to help improve theperformance of a free space or fiber communication or quantum computingsystem. Specifically, OAM states could be used as a different dimensionto encode bits on a single pulse (or a single photon), or be used tocreate additional data carriers in an SDM system.

There are some potential benefits of using OAM for communications andquantum computing, some specially designed novel fibers allow less modecoupling and cross talk while propagating in fibers. In addition, OAMbeams with different states share a ring-shaped beam profile, whichindicate rotational insensitivity for receiving the beams. Since thedistinction of OAM beams does not rely on the wavelength orpolarization, OAM multiplexing could be used in addition to WDM and PDMtechniques so that potentially improve the system performance may beprovided.

Referring now to the drawings, and more particularly to FIG. 8, whereinthere is illustrated two manners for increasing spectral efficiency of acommunications or quantum computing system. In general, there arebasically two ways to increase spectral efficiency 802 of acommunications or quantum computing system. The increase may be broughtabout by signal processing techniques 804 in the modulation scheme orusing multiple access technique. Additionally, the spectral efficiencycan be increase by creating new Eigen channels 806 within theelectromagnetic propagation. These two techniques are completelyindependent of one another and innovations from one class can be addedto innovations from the second class. Therefore, the combination of thistechnique introduced a further innovation.

Spectral efficiency 802 is the key driver of the business model of acommunications or quantum computing system. The spectral efficiency isdefined in units of bit/sec/hz and the higher the spectral efficiency,the better the business model. This is because spectral efficiency cantranslate to a greater number of users, higher throughput, higherquality or some of each within a communications or quantum computingsystem.

Regarding techniques using signal processing techniques or multipleaccess techniques. These techniques include innovations such as TDMA,FDMA, CDMA, EVDO, GSM, WCDMA, HSPA and the most recent OFDM techniquesused in 4G WIMAX and LTE. Almost all of these techniques use decades-oldmodulation techniques based on sinusoidal Eigen functions called QAMmodulation. Within the second class of techniques involving the creationof new Eigen channels 806, the innovations include diversity techniquesincluding space and polarization diversity as well as multipleinput/multiple output (MIMO) where uncorrelated radio paths createindependent Eigen channels and propagation of electromagnetic waves.

Referring now to FIG. 9, the present system configuration introduces twotechniques, one from the signal processing techniques 804 category andone from the creation of new eigen channels 806 category that areentirely independent from each other. Their combination provides aunique manner to disrupt the access part of an end to end communicationsor quantum computing system from twisted pair and cable to fiber optics,to free space optics, to RF used in cellular, backhaul and satellite, toRF satellite, to RF broadcast, to RF point-to point, to RFpoint-to-multipoint, to RF point-to-point (backhaul), to RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet), toInternet of Things (IOT), to Wi-Fi, to Bluetooth, to a personal devicecable replacement, to an RF and FSO hybrid system, to Radar, toelectromagnetic tags and to all types of wireless access. The firsttechnique involves the use of a new signal processing technique usingnew orthogonal signals to upgrade QAM modulation using non-sinusoidalfunctions. This is referred to as quantum level overlay (QLO) 902. Thesecond technique involves the application of new electromagneticwavefronts using a property of electromagnetic waves or photon, calledorbital angular momentum (QAM) 104. Application of each of the quantumlevel overlay techniques 902 and orbital angular momentum application904 uniquely offers orders of magnitude higher spectral efficiency 906within communication or quantum computing systems in their combination.

With respect to the quantum level overlay technique 909, new eigenfunctions are introduced that when overlapped (on top of one anotherwithin a symbol) significantly increases the spectral efficiency of thesystem. The quantum level overlay technique 902 borrows from quantummechanics, special orthogonal signals that reduce the time bandwidthproduct and thereby increase the spectral efficiency of the channel.Each orthogonal signal is overlaid within the symbol acts as anindependent channel. These independent channels differentiate thetechnique from existing modulation techniques.

With respect to the application of orbital angular momentum 904, thistechnique introduces twisted electromagnetic waves, or light beams,having helical wave fronts that carry orbital angular momentum (OAM).Different OAM carrying waves/beams can be mutually orthogonal to eachother within the spatial domain, allowing the waves/beams to beefficiently multiplexed and demultiplexed within a link. OAM beams areinteresting in communications or quantum computing due to theirpotential ability in special multiplexing multiple independent datacarrying channels.

With respect to the combination of quantum level overlay techniques 902and orbital angular momentum application 904, the combination is uniqueas the OAM multiplexing technique is compatible with otherelectromagnetic techniques such as wave length and polarization divisionmultiplexing. This suggests the possibility of further increasing systemperformance. The application of these techniques together in highcapacity data transmission disrupts the access part of an end to endcommunications or quantum computing system from twisted pair and cableto fiber optics, to free space optics, to RF used in cellular, backhauland satellite, to RF satellite, to RF broadcast, to RF point-to point,to RF point-to-multipoint, to RF point-to-point (backhaul), to RFpoint-to-point (fronthaul to provide higher throughput CPRI interfacefor cloudification and virtualization of RAN and cloudified HetNet), toInternet of Things (IOT), to Wi-Fi, to Bluetooth, to a personal devicecable replacement, to an RF and FSO hybrid system, to Radar, toelectromagnetic tags and to all types of wireless access.

Each of these techniques can be applied independent of one another, butthe combination provides a unique opportunity to not only increasespectral efficiency, but to increase spectral efficiency withoutsacrificing distance or signal to noise ratios.

Using the Shannon Capacity Equation, a determination may be made ifspectral efficiency is increased. This can be mathematically translatedto more bandwidth. Since bandwidth has a value, one can easily convertspectral efficiency gains to financial gains for the business impact ofusing higher spectral efficiency. Also, when sophisticated forward errorcorrection (FEC) techniques are used, the net impact is higher qualitybut with the sacrifice of some bandwidth. However, if one can achievehigher spectral efficiency (or more virtual bandwidth), one cansacrifice some of the gained bandwidth for FEC and therefore higherspectral efficiency can also translate to higher quality.

System providers are interested in increasing spectral efficiency.However, the issue with respect to this increase is the cost. Eachtechnique at different layers of the protocol has a different price tagassociated therewith. Techniques that are implemented at a physicallayer have the most impact as other techniques can be superimposed ontop of the lower layer techniques and thus increase the spectralefficiency further. The price tag for some of the techniques can bedrastic when one considers other associated costs. For example, themultiple input multiple output (MIMO) technique uses additional antennasto create additional paths where each RF path can be treated as anindependent channel and thus increase the aggregate spectral efficiency.In the MIMO scenario, the operator has other associated soft costsdealing with structural issues such as antenna installations, etc. Thesetechniques not only have tremendous cost, but they have huge timingissues as the structural activities take time and the achieving ofhigher spectral efficiency comes with significant delays which can alsobe translated to financial losses.

The quantum level overlay technique 902 has an advantage that theindependent channels are created within the symbols. This will have atremendous cost and time benefit compared to other techniques. Also, thequantum layer overlay technique 902 is a physical layer technique, whichmeans there are other techniques at higher layers of the protocol thatcan all ride on top of the QLO techniques 902 and thus increase thespectral efficiency even further. QLO technique 902 uses standard QAMmodulation used in OFDM based multiple access technologies such as WIMAXor LTE. QLO technique 902 basically enhances the QAM modulation at thetransceiver by injecting new signals to the I & Q components of thebaseband and overlaying them before QAM modulation as will be more fullydescribed herein below. At the receiver, the reverse procedure is usedto separate the overlaid signal and the net effect is a pulse shapingthat allows better localization of the spectrum compared to standard QAMor even the root raised cosine. The impact of this technique is asignificantly higher spectral efficiency.

Referring now more particularly to FIG. 10, there is illustrated ageneral overview of the manner for providing improved communicationand/or data transmission bandwidth within various interfaces 1002, usinga combination of multiple level overlay modulation 1004 and theapplication of orbital angular momentum 1006 to increase the number ofcommunications channels or amount of transmitted data.

The various interfaces 1002 may comprise a variety of links, such as RFcommunication, wireline communication such as cable or twisted pairconnections, or optical communications making use of light wavelengthssuch as fiber-optic communications or free-space optics. Various typesof RF communications may include a combination of RF microwave or RFsatellite communication, as well as multiplexing between RF andfree-space optics in real time.

By combining a multiple layer overlay modulation technique 1004 withorbital angular momentum (OAM) technique 1006, a higher throughput overvarious types of links 1002 may be achieved. The use of multiple leveloverlay modulation alone without OAM increases the spectral efficiencyof communication links 1002, whether wired, optical, or wireless.However, with OAM, the increase in spectral efficiency is even moresignificant.

Multiple overlay modulation techniques 1004 provide a new degree offreedom beyond the conventional 2 degrees of freedom, with time T andfrequency F being independent variables in a two-dimensional notationalspace defining orthogonal axes in an information diagram. This comprisesa more general approach rather than modeling signals as fixed in eitherthe frequency or time domain. Previous modeling methods using fixed timeor fixed frequency are considered to be more limiting cases of thegeneral approach of using multiple level overlay modulation 1004. Withinthe multiple level overlay modulation technique 1004, signals may bedifferentiated in two-dimensional space rather than along a single axis.Thus, the information-carrying capacity of a communications channel maybe determined by a number of signals which occupy different time andfrequency coordinates and may be differentiated in a notationaltwo-dimensional space.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals and/or data,with higher resulting information-carrying capacity, within an allocatedchannel. Given the frequency channel delta (Δf), a given signaltransmitted through it in minimum time Δt will have an envelopedescribed by certain time-bandwidth minimizing signals. Thetime-bandwidth products for these signals take the form:ΔtΔf=1/2(2n+1)where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms.

The orbital angular momentum process 1006 provides a twist to wavefronts of the electromagnetic fields carrying the data stream that mayenable the transmission of multiple data streams on the same frequency,wavelength, or other signal-supporting mechanism. Similarly, otherorthogonal signals may be applied to the different data streams toenable transmission of multiple data streams on the same frequency,wavelength or other signal-supporting mechanism. This will increase thebandwidth over a communications link by allowing a single frequency orwavelength to support multiple eigen channels, each of the individualchannels having a different orthogonal and independent orbital angularmomentum associated therewith.

Referring now more particularly to FIG. 11, there is illustrated afunctional block diagram of a system for generating the orbital angularmomentum “twist” within a communication or quantum computing system,such as that illustrated with respect to FIG. 10, to provide a datastream that may be combined with multiple other data streams fortransmission upon a same wavelength or frequency. Multiple data streams1102 are provided to the transmission processing circuitry 1100. Each ofthe data streams 1102 comprises, for example, an end to end linkconnection carrying a voice call or a packet connection transmittingnon-circuit switch packed data over a data connection. The multiple datastreams 1102 are processed by modulator/demodulator circuitry 1104. Themodulator/demodulator circuitry 1104 modulates the received data stream1102 onto a wavelength or frequency channel using a multiple leveloverlay modulation technique, as will be more fully described hereinbelow. The communications link may comprise an optical fiber link,free-space optics link, RF microwave link, RF satellite link, wired link(without the twist), etc.

The modulated data stream is provided to the orbital angular momentum(OAM) signal processing block 1106. The orbital angular momentum signalprocessing block 1106 applies in one embodiment an orbital angularmomentum to a signal. In other embodiments, the processing block 1106can apply any orthogonal function to a signal being transmitted. Theseorthogonal functions can be spatial Bessel functions, Laguerre-Gaussianfunctions, Hermite-Gaussian functions or any other orthogonal function.Each of the modulated data streams from the modulator/demodulator 1104are provided a different orbital angular momentum by the orbital angularmomentum electromagnetic block 1106 such that each of the modulated datastreams have a unique and different orbital angular momentum associatedtherewith. Each of the modulated signals having an associated orbitalangular momentum are provided to an optical transmitter 1108 thattransmits each of the modulated data streams having a unique orbitalangular momentum on a same wavelength. Each wavelength has a selectednumber of bandwidth slots B and may have its data transmissioncapability increase by a factor of the number of degrees of orbitalangular momentum

that are provided from the OAM electromagnetic block 1106. The opticaltransmitter 1108 transmitting signals at a single wavelength couldtransmit B groups of information. The optical transmitter 1108 and OAMelectromagnetic block 1106 may transmit

×B groups of information according to the configuration describedherein.

In a receiving mode, the optical transmitter 1108 will have a wavelengthincluding multiple signals transmitted therein having different orbitalangular momentum signals embedded therein. The optical transmitter 1108forwards these signals to the OAM signal processing block 1106, whichseparates each of the signals having different orbital angular momentumand provides the separated signals to the demodulator circuitry 1104.The demodulation process extracts the data streams 1102 from themodulated signals and provides it at the receiving end using themultiple layer overlay demodulation technique.

Referring now to FIG. 12, there is provided a more detailed functionaldescription of the OAM signal processing block 1106. Each of the inputdata streams are provided to OAM circuitry 1202. Each of the OAMcircuitry 1202 provides a different orbital angular momentum to thereceived data stream. The different orbital angular momentums areachieved by applying different currents for the generation of thesignals that are being transmitted to create a particular orbitalangular momentum associated therewith. The orbital angular momentumprovided by each of the OAM circuitries 1202 are unique to the datastream that is provided thereto. An infinite number of orbital angularmomentums may be applied to different input data streams or photonsusing many different currents. Each of the separately generated datastreams are provided to a signal combiner 1204, whichcombines/multiplexes the signals onto a wavelength for transmission fromthe transmitter 1206. The combiner 1204 performs a spatial mode divisionmultiplexing to place all of the signals upon a same carrier signal inthe space domain.

Referring now to FIG. 13, there is illustrated the manner in which theOAM processing circuitry 1106 may separate a received signal intomultiple data streams. The receiver 1302 receives the combined OAMsignals on a single wavelength and provides this information to a signalseparator 1304. The signal separator 1304 separates each of the signalshaving different orbital angular momentums from the received wavelengthand provides the separated signals to OAM de-twisting circuitry 1306.The OAM de-twisting circuitry 1306 removes the associated OAM twist fromeach of the associated signals and provides the received modulated datastream for further processing. The signal separator 1304 separates eachof the received signals that have had the orbital angular momentumremoved therefrom into individual received signals. The individuallyreceived signals are provided to the receiver 1302 for demodulationusing, for example, multiple level overlay demodulation as will be morefully described herein below.

FIG. 14 illustrates in a manner in which a single wavelength orfrequency, having two quanti-spin polarizations may provide an infinitenumber of twists having various orbital angular momentums associatedtherewith. The 1 axis represents the various quantized orbital angularmomentum states which may be applied to a particular signal at aselected frequency or wavelength. The symbol omega (o) represents thevarious frequencies to which the signals of differing orbital angularmomentum may be applied. The top grid 1402 represents the potentiallyavailable signals for a left handed signal polarization, while thebottom grid 1404 is for potentially available signals having righthanded polarization.

By applying different orbital angular momentum states to a signal at aparticular frequency or wavelength, a potentially infinite number ofstates may be provided at the frequency or wavelength. Thus, the stateat the frequency Δω or wavelength 1406 in both the left handedpolarization plane 1402 and the right handed polarization plane 1404 canprovide an infinite number of signals at different orbital angularmomentum states Δl. Blocks 1408 and 1410 represent a particular signalhaving an orbital angular momentum Δl at a frequency Δω or wavelength inboth the right handed polarization plane 1404 and left handedpolarization plane 1410, respectively. By changing to a differentorbital angular momentum within the same frequency Δω or wavelength1406, different signals may also be transmitted. Each angular momentumstate corresponds to a different determined current level fortransmission from the optical transmitter. By estimating the equivalentcurrent for generating a particular orbital angular momentum within theoptical domain and applying this current for transmission of thesignals, the transmission of the signal may be achieved at a desiredorbital angular momentum state.

Thus, the illustration of FIG. 15, illustrates two possible angularmomentums, the spin angular momentum, and the orbital angular momentum.The spin version is manifested within the polarizations of macroscopicelectromagnetism, and has only left and right hand polarizations due toup and down spin directions. However, the orbital angular momentumindicates an infinite number of states that are quantized. The paths aremore than two and can theoretically be infinite through the quantizedorbital angular momentum levels.

It is well-known that the concept of linear momentum is usuallyassociated with objects moving in a straight line. The object could alsocarry angular momentum if it has a rotational motion, such as spinning(i.e., spin angular momentum (SAM) 1502), or orbiting around an axis1506 (i.e., OAM 1504), as shown in FIGS. 15A and 15B, respectively. Alight beam may also have rotational motion as it propagates. In paraxialapproximation, a light beam carries SAM 1502 if the electrical fieldrotates along the beam axis 1506 (i.e., circularly polarized light1505), and carries OAM 1504 if the wave vector spirals around the beamaxis 1506, leading to a helical phase front 1508, as shown in FIGS. 15Cand 15D. In its analytical expression, this helical phase front 1508 isusually related to a phase term of exp(i

θ) in the transverse plane, where θ refers to the angular coordinate,and

is an integer indicating the number of intertwined helices (i.e., thenumber of 2π phase shifts along the circle around the beam axis).

could be a positive, negative integer or zero, corresponding toclockwise, counterclockwise phase helices or a Gaussian beam with nohelix, respectively.

Two important concepts relating to OAM include: 1) OAM and polarization:As mentioned above, an OAM beam is manifested as a beam with a helicalphase front and therefore a twisting wave vector, while polarizationstates can only be connected to SAM 1502. A light beam carries SAM 1502of ±h/2π (h is Plank's constant) per photon if it is left or rightcircularly polarized, and carries no SAM 1502 if it is linearlypolarized. Although the SAM 1502 and OAM 1504 of light can be coupled toeach other under certain scenarios, they can be clearly distinguishedfor a paraxial light beam. Therefore, with the paraxial assumption, OAM1504 and polarization can be considered as two independent properties oflight. 2) OAM beam and Laguerre-Gaussian (LG) beam: In general, anOAM-carrying beam could refer to any helically phased light beam,irrespective of its radial distribution (although sometimes OAM couldalso be carried by a non-helically phased beam). An LG beam is a specialsubset among all OAM-carrying beams, due to that the analyticalexpression of LG beams are eigen-solutions of paraxial form of the waveequation in a cylindrical coordinates. For an LG beam, both azimuthaland radial wavefront distributions are well defined, and are indicatedby two index numbers,

and p, of which

has the same meaning as that of a general OAM beam, and p refers to theradial nodes in the intensity distribution. Mathematical expressions ofLG beams form an orthogonal and complete basis in the spatial domain. Incontrast, a general OAM beam actually comprises a group of LG beams(each with the same

index but a different p index) due to the absence of radial definition.The

term of “OAM beam” refers to all helically phased beams, and is used todistinguish from LG beams.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the radiationtransmitted by the signals. The Maxwell-Heaviside equations can berepresented as:

${\nabla{\cdot E}} = \frac{\rho}{ɛ_{0}}$${\nabla{\times E}} = {- \frac{\partial B}{\partial t}}$ ∇⋅B = 0${\nabla{\times B}} = {{ɛ_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,\ x} \right)}}}$

where ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, one can derive 23symmetries/conserved quantities from Maxwell's original equations.However, there are only ten well-known conserved quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

Maxwell's linear theory is of U(1) symmetry with Abelian commutationrelations. They can be extended to higher symmetry group SU(2) form withnon-Abelian commutation relations that address global (non-local inspace) properties. The Wu-Yang and Harmuth interpretation of Maxwell'stheory implicates the existence of magnetic monopoles and magneticcharges. As far as the classical fields are concerned, these theoreticalconstructs are pseudo-particle, or instanton. The interpretation ofMaxwell's work actually departs in a significant ways from Maxwell'soriginal intention. In Maxwell's original formulation, Faraday'selectrotonic states (the Aμ field) was central making them compatiblewith Yang-Mills theory (prior to Heaviside). The mathematical dynamicentities called solitons can be either classical or quantum, linear ornon-linear and describe EM waves. However, solitons are of SU(2)symmetry forms. In order for conventional interpreted classicalMaxwell's theory of U(1) symmetry to describe such entities, the theorymust be extended to SU(2) forms.

Besides the half dozen physical phenomena (that cannot be explained withconventional Maxwell's theory), the recently formulated Harmuth Ansatzalso address the incompleteness of Maxwell's theory. Harmuth amendedMaxwell's equations can be used to calculate EM signal velocitiesprovided that a magnetic current density and magnetic charge are addedwhich is consistent to Yang-Mills filed equations. Therefore, with thecorrect geometry and topology, the Aμ potentials always have physicalmeaning

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

 Hamiltonian  (total  energy)$H = {{\sum\limits_{i}{m_{i}\gamma_{i}c^{2}}} + {\frac{ɛ_{0}}{2}{\int{d^{3}{x\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}}$ conservation  of  energy${\frac{dU^{mech}}{dt} + \frac{dU^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot S}}}} = 0$

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

linear  momentum$p = {{\sum\limits_{i}{m_{i}\gamma_{i}v_{i}}} + {ɛ_{0}{\int{d^{3}{x\left( {E \times B} \right)}}}}}$conservation  of  linear  momentum${\frac{dp^{mech}}{dt} + \frac{dp^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot T}}}} = 0$

The conservation of system center of energy is represented by theequation:

$R = {{\frac{1}{H}{\sum\limits_{i}{\left( {x_{i} - x_{0}} \right)m_{i}\gamma_{i}c^{2}}}} + {\frac{ɛ_{0}}{2H}{\int{d^{3}{x\left( {x - x_{0}} \right)}\;\left( {{E^{2}} + {c^{2}{B^{2}}}} \right)}}}}$

Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

conservation  of  angular  momentum${\frac{dJ^{mech}}{dt} + \frac{dJ^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot M}}}} = 0$

For radiation beams in free space, the EM field angular momentum J^(em)can be separated into two parts:J ^(em)=ε₀∫_(V′) d ³ x′(E×A)+ε₀∫_(V′) d ³ x′E _(i)[(x′−x ₀)×∇]A _(i)

For each singular Fourier mode in real valued representation:

$J^{em} = {{{- i}\frac{ɛ_{0}}{2\;\omega}{\int_{V^{\prime}}{d^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {i\frac{ɛ_{0}}{2\;\omega}{\int_{V^{\prime}}{d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}$

The first part is the EM spin angular momentum S^(em), its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum L^(em) its classical manifestation is wavehelicity. In general, both EM linear momentum P^(em), and EM angularmomentum J^(em)=L^(em)+S^(em) are radiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

$\begin{matrix}{\;{{{\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0},}} & {{continuity}\mspace{14mu}{equation}}\end{matrix}$where S is the Poynting vectorS=¼(E×H*+E*×H),and U is the energy densityU=¼(ε|E| ²+μ₀ |H| ²),with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$V = {{\nabla \times v_{opt}} = {\nabla \times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}}$

                            Maxwell′s  Equations ${\begin{matrix}{{Gauss}'} \\{Laws} \\{{{Faraday}'}s} \\{Law} \\{{{Ampere}'}s} \\{Law}\end{matrix}\mspace{31mu}\left. \begin{matrix}{{\nabla{\cdot D}} = \rho} \\{{\nabla{\cdot B}} = 0} \\\; \\{{\nabla{\times E}} = {- \frac{\partial B}{\partial t}}} \\{{\nabla{\times H}} = {J + \frac{\partial D}{\partial t}}}\end{matrix}\mspace{130mu}\downarrow\begin{matrix}{{{\nabla^{2}E} + {k^{2}E}} = 0}\end{matrix} \right.\mspace{290mu}\left. \left( {{Full}\mspace{14mu}{Wave}\mspace{14mu}{Equation}} \right)\mspace{130mu}\downarrow\mspace{11mu}\mspace{484mu}{Wave} \right.\mspace{14mu}{Equations}}\mspace{40mu}$$\begin{matrix}{{\frac{d^{2}E}{dx^{2}} + \frac{d^{2}E}{dy^{2}} + \frac{d^{2}E}{dz^{2}} + {k^{2}E}} = 0} \\{{{\frac{1}{\rho}\frac{d}{d\;\rho}\left( {\rho\frac{d}{d\;\rho}} \right)E} + {\frac{1}{\rho^{2}}\frac{d^{2}}{d\;\varphi^{2}}E} + \frac{d^{2}E}{{dz}^{2}} + {k^{2}E}} = 0}\end{matrix}\mspace{25mu}\begin{matrix}\begin{matrix}({Rectangular}) \\\;\end{matrix} \\({Cylindrical})\end{matrix}$

Referring now to FIGS. 16A and 16B, there is illustrated the manner inwhich a signal and its associated Poynting vector in a plane wavesituation. In the plane wave situation illustrated generally at 1602,the transmitted signal may take one of three configurations. When theelectric field vectors are in the same direction, a linear signal isprovided, as illustrated generally at 1604. Within a circularpolarization 1606, the electric field vectors rotate with the samemagnitude. Within the elliptical polarization 1608, the electric fieldvectors rotate but have differing magnitudes. The Poynting vectorremains in a constant direction for the signal configuration to FIG. 16Aand always perpendicular to the electric and magnetic fields. Referringnow to FIG. 16B, when a unique orbital angular momentum is applied to asignal as described here and above, the Poynting vector S 1610 willspiral about the direction of propagation of the signal. This spiral maybe varied in order to enable signals to be transmitted on the samefrequency as described herein.

FIGS. 17A through 17C illustrate the differences in signals havingdifferent helicity (i.e., orbital angular momentums). Each of thespiraling Poynting vectors associated with the signals 1702, 1704, and1706 provide a different shaped signal. Signal 1702 has an orbitalangular momentum of +1, signal 1704 has an orbital angular momentum of+3, and signal 1706 has an orbital angular momentum of −4. Each signalhas a distinct angular momentum and associated Poynting vector enablingthe signal to be distinguished from other signals within a samefrequency. This allows differing type of information to be transmittedon the same frequency, since these signals are separately detectable anddo not interfere with each other (Eigen channels).

FIG. 17D illustrates the propagation of Poynting vectors for variousEigen modes. Each of the rings 1720 represents a different Eigen mode ortwist representing a different orbital angular momentum within the samefrequency. Each of these rings 1720 represents a different orthogonalchannel. Each of the Eigen modes has a Poynting vector 1722 associatedtherewith.

Topological charge may be multiplexed to the frequency for either linearor circular polarization. In case of linear polarizations, topologicalcharge would be multiplexed on vertical and horizontal polarization. Incase of circular polarization, topological charge would multiplex onleft hand and right hand circular polarizations. The topological chargeis another name for the helicity index “I” or the amount of twist or OAMapplied to the signal. Also, use of the orthogonal functions discussedherein above may also be multiplexed together onto a same signal inorder to transmit multiple streams of information. The helicity indexmay be positive or negative. In wireless communications, differenttopological charges/orthogonal functions can be created and muxedtogether and de-muxed to separate the topological chargescharges/orthogonal functions. The signals having different orthogonalfunction are spatially combined together on a same signal but do notinterfere with each other since they are orthogonal to each other.

The topological charges

s can be created using Spiral Phase Plates (SPPs) as shown in FIG. 17Eusing a proper material with specific index of refraction and ability tomachine shop or phase mask, holograms created of new materials or a newtechnique to create an RF version of Spatial Light Modulator (SLM) thatdoes the twist of the RF waves (as opposed to optical beams) byadjusting voltages on the device resulting in twisting of the RF waveswith a specific topological charge. Spiral Phase plates can transform aRF plane wave (

=0) to a twisted RF wave of a specific helicity (i.e.

=+1).

Cross talk and multipath interference can be corrected using RFMultiple-Input-Multiple-Output (MIMO). Most of the channel impairmentscan be detected using a control or pilot channel and be corrected usingalgorithmic techniques (closed loop control system).

While the application of orbital angular momentum to various signalsallow the signals to be orthogonal to each other and used on a samesignal carrying medium, other orthogonal function/signals can be appliedto data streams to create the orthogonal signals on the same signalmedia carrier.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity, within an allocated channel.Given the frequency channel delta (Δf), a given signal transmittedthrough it in minimum time Δt will have an envelope described by certaintime-bandwidth minimizing signals. The time-bandwidth products for thesesignals take the form:ΔtΔf=½(2n+1)where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms. These types of orthogonal signals that reduce the timebandwidth product and thereby increase the spectral efficiency of thechannel.

Hermite-Gaussian polynomials are one example of a classical orthogonalpolynomial sequence, which are the Eigenstates of a quantum harmonicoscillator. Signals based on Hermite-Gaussian polynomials possess theminimal time-bandwidth product property described above, and may be usedfor embodiments of systems. However, it should be understood that othersignals may also be used, for example orthogonal polynomials such asJacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials, and Laguerre-Gaussian polynomials. Q-functionsare another class of functions that can be employed as a basis for MLOsignals.

In addition to the time bandwidth minimization described above, theplurality of data streams can be processed to provide minimization ofthe Space-Momentum products in spatial modulation. In this case:

${\Delta x\Delta p} = \frac{1}{2}$

Processing of the data streams in this manner create wavefronts that arespatial. The processing creates wavefronts that are also orthogonal toeach other like the OAM twisted functions but these comprise differenttypes of orthogonal functions that are in the spatial domain rather thanthe temporal domain.

The above described scheme is applicable to twisted pair, coaxial cable,fiber optic, RF satellite, RF broadcast, RF point-to point, RFpoint-to-multipoint, RF point-to-point (backhaul), RF point-to-point(fronthaul to provide higher throughput CPRI interface forcloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE.

Hermite Gaussian Beams

Hermite Gaussian beams may also be used for transmitting orthogonal datastreams. In the scalar field approximation (e.g. neglecting the vectorcharacter of the electromagnetic field), any electric field amplitudedistribution can be represented as a superposition of plane waves, i.e.by:

$E \propto {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\;\pi} \right)^{2}}{A\left( {k_{x},k_{y}} \right)}e^{{{ik}_{x}x} + {{ik}_{y}y} + {{ik}_{z}z} + {iz}}\sqrt{k^{2} - k_{x}^{2} - k_{y}^{2}}}}}$

This representation is also called angular spectrum of plane waves orplane-wave expansion of the electromagnetic field. Here A(k_(x), k_(y))is the amplitude of the plane wave. This representation is chosen insuch a way that the net energy flux connected with the electromagneticfield is towards the propagation axis z. Every plane wave is connectedwith an energy flow that has direction k. Actual lasers generate aspatially coherent electromagnetic field which has a finite transversalextension and propagates with moderate spreading. That means that thewave amplitude changes only slowly along the propagation axis (z-axis)compared to the wavelength and finite width of the beam. Thus, theparaxial approximation can be applied, assuming that the amplitudefunction A(k_(x), k_(y)) falls off sufficiently fast with increasingvalues of (k_(x), k_(y)).

Two principal characteristics of the total energy flux can beconsidered: the divergence (spread of the plane wave amplitudes in wavevector space), defined as:

${Divergence} \propto {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\;\pi} \right)^{2}}\left( {K_{x}^{2} + K_{y}^{2}} \right){{A\left( {k_{x},k_{y}} \right)}}^{2}}}}$

and the transversal spatial extension (spread of the field intensityperpendicular to the z-direction) defined as:

${{{Transversal}\mspace{14mu}{Extention}} \propto {\int_{- \infty}^{\infty}{{dx}{\int_{- \infty}^{\infty}{{{dy}\left( {x^{2} + y^{2}} \right)}{E}^{2}}}}}} = {\int{\int{\frac{{dk}_{x}{dk}_{y}}{\left( {2\;\pi} \right)^{2}}\left\lbrack {{\frac{\partial A}{\partial x}}^{2} + {\frac{\partial A}{\partial y}}^{2}} \right\rbrack}}}$

Let's now look for the fundamental mode of the beam as theelectromagnetic field having simultaneously minimal divergence andminimal transversal extension, i.e. as the field that minimizes theproduct of divergence and extension. By symmetry reasons, this leads tolooking for an amplitude function minimizing the product:

${\left\lbrack {\int_{- \infty}^{\infty}{\frac{{dk}_{x}}{\left( {2\;\pi} \right)}k_{x}^{2}{A}^{2}}} \right\rbrack\;\left\lbrack {\int_{- \infty}^{\infty}{\frac{{dk}_{x}}{\left( {2\;\pi} \right)}{\frac{\partial A}{\partial k_{x}}}^{2}}} \right\rbrack} = \frac{{A}^{4}}{\left( {8\;\pi^{2}} \right)^{2}}$

Thus, seeking the field with minimal divergence and minimal transversalextension can lead directly to the fundamental Gaussian beam. This meansthat the Gaussian beam is the mode with minimum uncertainty, i.e. theproduct of its sizes in real space and wave-vector space is thetheoretical minimum as given by the Heisenberg's uncertainty principleof Quantum Mechanics. Consequently, the Gaussian mode has lessdispersion than any other optical field of the same size, and itsdiffraction sets a lower threshold for the diffraction of real opticalbeams.

Hermite-Gaussian beams are a family of structurally stable laser modeswhich have rectangular symmetry along the propagation axis. In order toderive such modes, the simplest approach is to include an additionalmodulation of the form:

$E_{m,n}^{H} = {\int_{- \infty}^{\infty}{\frac{{dk}_{x}{dk}_{y}}{\left( {2\;\pi} \right)^{2}}\left( {ik}_{x} \right)^{m}\left( {ik}_{y} \right)^{n}e^{S}}}$${S\left( {k_{x},k_{y},x,y,z} \right)} = {{{ik}_{x}x} + {{ik}_{y}y} + {{ik}_{z}z} - {\frac{W_{0}}{4}{\left( {1 + {i\frac{Z}{Z_{R}}}} \right)\left\lbrack {k_{x}^{2} + k_{y}^{2}} \right\rbrack}}}$

The new field modes occur to be differential derivatives of thefundamental Gaussian mode E₀.

$E_{m,n}^{H} = {\frac{\partial^{m + n}}{{\partial x^{m}}{\partial y^{n}}}E_{0}}$

Looking at the explicit form E0 shows that the differentiations in thelast equation lead to expressions of the form:

$\frac{\partial^{P}}{\partial x^{p}}e^{({{- \alpha}\; x^{2}})}$with some constant p and α. Using now the definition of Hermits'polynomials,

${H_{p}(x)} = {\left( {- 1} \right)^{p}e^{(x^{2})}\frac{d^{P}}{{dx}^{p}}e^{({{- \alpha}\; x^{2}})}}$

Then the field amplitude becomes

${E_{m,n}^{H}\left( {x,y,z} \right)} = {\sum\limits_{m}{\sum\limits_{n}{C_{mn}E_{0}\frac{w_{0}}{w(z)}{H_{m}\left( {\sqrt{2}\frac{x}{w(z)}} \right)}{H_{n}\left( {\sqrt{2}\frac{y}{w(z)}} \right)}e^{\frac{- {({x^{2} + y^{2}})}}{{w{(z)}}^{2}}}e^{{- {j{({m + n + 1})}}}\tan^{- 1}{z/z_{R}}}e^{\frac{- {({x^{2} + y^{2}})}}{2{R{(z)}}}}}}}$Where

${\rho^{2} = {x^{2} + y^{2}}}{\xi = \frac{z}{z_{R}}}$and Rayleigh length z_(R)

$z_{R} = \frac{\pi w_{0}^{2}}{\lambda}$And beam diameterw(ξ)=w ₀√{square root over ((1+μ²))}

In cylindrical coordinates, the field takes the form:

${E_{l,p}^{L}\left( {\rho,\varphi,z} \right)} = {\sum\limits_{l}{\sum\limits_{np}{C_{lp}E_{0}\frac{w_{0}}{w(z)}\left( {\sqrt{2}\frac{\rho}{w(z)}} \right)^{l}{L_{p}^{l}\left( {\sqrt{2}\frac{\rho}{w(z)}} \right)}e^{\frac{- \rho^{2}}{{w{(z)}}^{2}}}e^{{- {j{({{2\; p} + l + 1})}}}\tan^{- 1}{z/z_{R}}}e^{{jl}\;\varphi}e^{\frac{{- {jk}}\;\rho^{2}}{2{R{(z)}}}}}}}$Where L_(p) ^(l) is Laguerre functions.

Mode division multiplexing (MDM) of multiple orthogonal beams increasesthe system capacity and spectral efficiency in optical communicationsystems. For free space systems, multiple beams each on a differentorthogonal mode can be transmitted through a single transmitter andreceiver aperture pair. Moreover, the modal orthogonality of differentbeans enables the efficient multiplexing at the transmitter anddemultiplexing at the receiver.

Different optical modal basis sets exist that exhibit orthogonality. Forexample, orbital angular momentum (OAM) beams that are either LaguerreGaussian (LG or Laguerre Gaussian light modes may be used formultiplexing of multiple orthogonal beams in free space optical and RFtransmission systems. However, there exist other modal groups that alsomay be used for multiplexing that do not contain OAM. Hermite Gaussian(HG) modes are one such modal group. The intensity of an HG_(m,n) beamis shown according to the equation:

${{I\left( {x,y,z} \right)} = {C_{m,n}{H_{m}^{2}\left( \frac{\sqrt{2}x}{w(z)} \right)}{H_{n}^{2}\left( \frac{\sqrt{2}y}{w(z)} \right)} \times {\exp\left( {{- \frac{2\; x^{2}}{{w(z)}^{2}}} - \frac{2y^{2}}{{w(z)}^{2}}} \right)}}},{{w(z)} = {w_{0}\sqrt{1 + \left\lbrack {\lambda\;{z/\pi}\; w_{0}^{2}} \right\rbrack}}}$

in which H_(m)(*) and H_(n)(*) are the Hermite polynomials of the mthand nth order. The value w₀ is the beam waist at distance Z=0. Thespatial orthogonality of HG modes with the same beam waist w₀ relies onthe orthogonality of Hermite polynomial in x or y directions.

Referring now to FIG. 18, there is illustrated a system for using theorthogonality of an HG modal group for free space spatial multiplexingin free space. A laser 1802 is provided to a beam splitter 1804. Thebeam splitter 1804 splits the beam into multiple beams that are eachprovided to a modulator 1806 for modulation with a data stream 1808. Themodulated beam is provided to collimators 1810 that provides acollimated light beam to spatial light modulators 1812. Spatial lightmodulators (SLM's) 1812 may be used for transforming input plane wavesinto HG modes of different orders, each mode carrying an independentdata channel. These HG modes are spatially multiplexed using amultiplexer 1814 and coaxially transmitted over a free space link 1816.At the receiver 1818 there are several factors that may affect thedemultiplexing of these HG modes, such as receiver aperture size,receiver lateral displacement and receiver angular error. These factorsaffect the performance of the data channel such as signal-to-noise ratioand crosstalk.

With respect to the characteristics of a diverged HG_(m,0) beam (m=0-6),the wavelength is assumed to be 1550 nm and the transmitted power foreach mode is 0 dBm. Higher order HG modes have been shown to have largerbeam sizes. For smaller aperture sizes less power is received for higherorder HG modes due to divergence.

Since the orthogonality of HG modes relies on the optical fielddistribution in the x and y directions, a finite receiver aperture maytruncate the beam. The truncation will destroy the orthogonality andcost crosstalk of the HG channels. When an aperture is smaller, there ishigher crosstalk to the other modes. When a finite receiver is used, ifan HG mode with an even (odd) order is transmitted, it only causes crosstalk to other HG modes with even (odd) numbers. This is explained by thefact that the orthogonality of the odd and even HG modal groups remainswhen the beam is systematically truncated.

Moreover, misalignment of the receiver may cause crosstalk. In oneexample, lateral displacement can be caused when the receiver is notaligned with the beam axis. In another example, angular error may becaused when the receiver is on axis but there is an angle between thereceiver orientation and the beam propagation axis. As the lateraldisplacement increases, less power is received from the transmittedpower mode and more power is leaked to the other modes. There is lesscrosstalk for the modes with larger mode index spacing from thetransmitted mode.

Mode Conversion Approaches

Referring now to FIG. 19, among all external-cavity methods, perhaps themost straightforward one is to pass a Gaussian beam through a coaxiallyplaced spiral phase plate (SPP) 1902. An SPP 1902 is an optical elementwith a helical surface, as shown in FIG. 17E. To produce an OAM beamwith a state of

, the thickness profile of the plate should be machined as

λθ/2π(n−1), where n is the refractive index of the medium. A limitationof using an SPP 1902 is that each OAM state requires a differentspecific plate. As an alternative, reconfigurable diffractive opticalelements, e.g., a pixelated spatial light modulator (SLM) 1904, or adigital micro-mirror device can be programmed to function as anyrefractive element of choice at a given wavelength. As mentioned above,a helical phase profile exp(i

θ) converts a linearly polarized Gaussian laser beam into an OAM mode,whose wave front resembles an

-fold corkscrew 1906, as shown at 1904. Importantly, the generated OAMbeam can be easily changed by simply updating the hologram loaded on theSLM 1904. To spatially separate the phase-modulated beam from thezeroth-order non-phase-modulated reflection from the SLM, a linear phaseramp is added to helical phase code (i.e., a “fork”-like phase pattern1908 to produce a spatially distinct first-order diffracted OAM beam,carrying the desired charge. It should also be noted that theaforementioned methods produce OAM beams with only an azimuthal indexcontrol. To generate a pure LG_(l,p) mode, one must jointly control boththe phase and the intensity of the wavefront. This could be achievedusing a phase-only SLM with a more complex phase hologram.

OAM Generation with DLP

The OAM signals used within quantum computers as described above may begenerated using Digital Light Processors (DLP). DLP comprises digitallight processors that are display devices based on opticalmicro-electro-mechanical technology that uses a digital micromirrordevice using for example technologies disclosed in U.S. patentapplication Ser. No. 14/864,511, entitled SYSTEM AND METHOD FOR APPLYINGORTHOGONAL LIMITATIONS TO LIGHT BEAMS USING MICROELECTROMECHANICALSYSTEMS, which is incorporated herein by reference in its entirety.Referring now to FIG. 20, there is illustrated a further manner forgenerating a light beam 2002 including orthogonal functions such as OAM,Hermite Gaussian, Laguerre Gaussian, etc., therein to encode informationin the beam. The laser beam generator 2004 generates a beam 2006including plane waves that is provided to a MicroElectroMechanicalsystem (MEMs) device 2008. Examples of MEMs devices 2008 include digitallight processing (DLP) projectors or digital micro-mirror devices (DMDs)that enable the generation of light beams having variouscharacteristics. A MEMs device 2008 can generate Hermite Gaussian (HG)modes, Laguerre Gaussian (LG) modes and vortex OAM modes that areprogrammed responsive to inputs to the MEMs device 2008. The MEMs device2008 has mode selection logic 2010 that enable selection of the LaguerreGaussian, Hermite Gaussian and vortex OAM modes (or other orthogonalfunction modes) for processing of the incoming light beam 2006. The MEMsdevice 2008 further enables switching between the different modes at avery high rate of a few thousand times per second which is notachievable using spatial light modulator (SLMs). Switching between themodes is controlled via mode switching logic 2012. This fast switchingenables these forms of OAM, HG or LG mode generation for communicationsas well as quantum key distribution (QKD) and quantum computers forquantum information processing. The orthogonal characteristics ofLaguerre-Gaussian (LG) with OAM and Hermite-Gaussian (HG) beams combinedwith high-speed switching of MEMs make the device useful in achievinghigher data capacity. This is possible using holograms that areprogrammed into the memory of a DLP that program micro-mirrors toselected positions and can twist a light beam with programmedinformation using the mirrors.

This enables the on-demand realization of binary gratings (holograms)that can be switched between at very high speed using an externaldigital signal. Using, for example, DLP technologies, a switch betweendifferent modes (different binary gratings) may be achieved at a veryhigh rate of speed of a few thousand times per second which is notachievable using spatial light modulators (SLMs). This allows for thedynamic control of helicities provided to a beam of light for a newmodulation and/or multiple access technique to encode information.

DLP's allow for high resolution and accuracy from micrometers tomillimeters thus enabling a variety of frequencies from infrared toultraviolet to be utilized. The use of DLP's for MDM (mode divisionmultiplexing) minimizes color, distance, movement and environmentalsensitivity and is thus ideal for building integrated optics. Themajority of SLM's are limited by a frame refresh rate of about 60 Hzwhich makes the high speed, wide range of operational spectral bandwidthof digital micro-mirror devices (DMD's) useful in a variety ofapplications. DMD designs inherently minimize temperature sensitivityfor reliable 3-D wave construction.

The vast majority of commercially available SLM devices are limited toframe rate of about 60 Hz which considerably limits the speed ofoperation of any system based on this technology. A DMD is an amplitudeonly spatial light modulator. The high speed, wide range of operationalspectral bandwidth and high power threshold of a DMDs makes the device auseful tool for variety of applications. Variations of DMD's arecommercially available for a fraction of the cost of a phase only SLM.Intensity shaping of spatial modes can be achieved by switching themicro mirrors on and off rapidly. However, the modes created during thisprocess may not be temporally stable and have the desired intensityprofile only when averaged by a slow detector.

Phase and amplitude information may be encoded by modulating theposition and width of a binary amplitude grating implemented within ahologram such as those illustrated in FIGS. 21A-21H. By implementingsuch holograms to control a DMD, HG modes, LG modes, OAM vortex modes orany angular (ANG) mode may be created by properly programming the DMDwith a hologram. Additionally, the switching between the generated modesmay be performed at a very high speed.

This approach may be realized by considering a one-dimensional binaryamplitude grating. The transmission function for this grating can bewritten as:

${\tau(x)} = {\sum\limits_{n = {- \infty}}^{\infty}{\prod\left\lbrack \frac{x - {\left( {n + k} \right)x_{0}}}{wx_{0}} \right\rbrack}}$where ${\prod(\nu)} = {{Rec{t(\nu)}} = \begin{Bmatrix}1 & {if} & {{\nu } \leq 1} \\0 & \; & {else}\end{Bmatrix}}$

This function can be pictured as a pulse train with a period of x₀. Theparameters of “k” and “w” are unitless quantities that set the positionand the width of each pulse and are equal to constant values for auniform grating. It is possible to locally change the value of theseparameters to achieve phase and amplitude modulations of the opticalfield. The transmittance function τ(x) is a periodic function and can beexpanded as a Fourier series.

In a case where k(x) and w(x) are functions of x and the binary gratingis illuminated by a monochromatic plane wave. The first order diffractedlight can be written as:

${\tau_{1}(x)} = {\frac{1}{\pi}{\sin\left\lbrack {\pi{w(x)}} \right\rbrack}e^{i\; 2\pi\;{k{(x)}}}}$

Thus, w(x) is related to the amplitude of the diffracted light whilek(x) sets its phase. Therefore, the phase and the amplitude of thediffracted light can be controlled by setting the parameters k(x) andw(x). In communication theory, these methods are sometimes referred toas pulse position modulation (PPM) and pulse width modulation (PWM). Theequation above is a good approximation for slowly varying k(x) and w(x)functions.

The above analysis treats a one-dimensional case. A two dimensionalgrating can be generated by thresholding a rapidly varying modulatedcarrier as:τ(x,y)=½+½ sgn{cos[2πx/x ₀ +πk(x,y)]−cos[πw(x,y)]}

Here, sgn(x, y) is the sign function. This may be checked in the limitwhere w(x,y) and k(x,y). One can find the corresponding w(x,y) andk(x,y) functions for a general complex scalar field:scaler field=A(x,y)e ^(iφ(x,y))According to the relations

${w\left( {x,y} \right)} = {\frac{1}{\pi}si{n^{- 1}\left\lbrack {{{A\left( {x,y} \right)}{k\left( {x,y} \right)}} = {\frac{1}{\pi}{\varphi\left( {x,y} \right)}}} \right.}}$

One could design 2-D binary amplitude holograms to generate LG modes.The gratings holograms designed for vortex modes would have a fairlyuniform width across the aperture whereas for the case of LG modes, thegratings gradually disappear when the amplitude gets negligibly small.

A digital micro-mirror device (DMD) is an amplitude only spatial lightmodulator. The device consist of an array of micro mirrors that can becontrolled in a binary fashion by setting the deflection angle of anindividual mirror to either +12° or −12°. Referring now to FIG. 22A,there is illustrated a general block diagram of a DMD 2202. The DMD 2202includes a plurality of micro-mirrors 2208 arranged in an X by Y array.The array may comprise a 1024×768 array of aluminum micro-mirrors suchas that implemented in the DLP 5500 DMD Array. However, it will beappreciated that other array sizes and DMD devices may be used. Eachmicro-mirror 2208 includes a combination of opto-mechanical andelectro-mechanical elements. Each micro-mirror 2208 comprises a pixel ofthe DMD 2202. The micro-mirror 2208 is an electromechanical elementhaving two stable micro-mirror states of +120 and −12°. Themicro-mirrors have a 10.8 micrometer pitch and are designed for lighthaving a wavelength of 420 nm-700 nm. The state of the micro-mirror 2208is determined by the geometry and electrostatics of the pixel duringoperation. The two positions of the micro-mirror 2208 determine thedirection that the light beam striking the mirror is deflected. Inparticular, the DMD 2202 is a spatial light modulator. By convention,the positive (+) state is tilted toward the illumination and is referredto as the “on” state. Similarly, the negative (−) state is tilted awayfrom the illumination and is referred to as the “off” state.

FIG. 22B illustrates the manner in which a micro-mirror 2208 willinteract with a light source 2230 such as a laser. The light source 2230shines a beam along angle of −24° that strikes the micro-mirror 2208.When the mirror is in the “off” state 2232 at an angle of −12°, the offstate energy 2234 is reflected at an angle of 48°. When the mirror 2208is positioned at the flat state 2236 of 0°, the flat state energy 2238is reflected in an angle of 24°. Finally, when the mirror is at +12° inthe “on” state 2240, the on state energy 2242 is reflected at 0° throughthe projection lens 2234 of a DMD.

Referring now to FIG. 23, there is illustrated a view of the mechanicalstructure of a micro-mirror 2208. The micro-mirror 2208 includes themirror 2302 attached to a torsional hinge 2304 along a diagonal axis2306 of the mirror. The underside of the micro-mirror 2302 makeselectrical contact with the remainder of the circuitry via spring tips2308. A pair of electrodes 2310 is used for holding the micro-mirror2302 in one of the two operational positions (+12° and −12°).

Referring now also to FIG. 24, there is illustrated a block diagram ofthe functional components of the micro-mirror 2208. Below eachmicro-mirror 2208 is a memory cell 2402 consisting of dual CMOS memoryelements 2404. The states of the two memory elements 2404 are notindependent, but are always complementary. If one CMOS memory element2404 is at a logical “1” level, the other CMOS element is at a logical“0” and vice versa. The state of the memory cell 2402 of themicro-mirror 2208 plays a part in the mechanical position of the mirror2208. However, loading information within the memory cell 2402 does notautomatically change the mechanical state of the micro-mirror 2208.

Although the state of the dual CMOS memory elements 2404 plays a part indetermining the state of the micro-mirror 2208, the state of the memoryelements 2304 is not the sole determining factor. Once the micro-mirror2208 has landed, changing the state of the memory cells 2402 will notcause the micro-mirror 2208 to flip to the other state. Thus, the memorystate and the micro-mirror state are not directly linked together. Inorder for the state of the CMOS memory elements 2404 to be transferredto the mechanical position of the micro-mirror 2208, the micro-mirror3108 must receive a “Mirror Clocking Pulse” signal. The mirror clockingpulse signal momentarily releases the micro-mirror 3108 and causes themirror to reposition based on the state of the CMOS memory elements2304. Thus, information relating to mirror positions may be preloadedinto the memory element 2404, and the mechanical position of the mirror2302 for each mirror within a MEMs device 2202 simultaneously changeresponsive to the mirror clocking pulse signal. One manner in which theinformation within the memory cells 2402 may be programmed is throughthe use of holograms, such as those described herein that are used todefined the position of each of the micro-mirrors 2208 with and a MEMsdevice 2202.

When a DMD 2202 is “powered up” or “powered down,” there are prescribedoperations that are necessary to ensure the proper orientation of themicro-mirrors 2208. These operations position the micro-mirrors 2208during power up and release them during power down. The process forchanging the position of a micro-mirror 2208 is more particularlyillustrated in the flowchart of FIG. 25. Initially, at step 2502, thememory states within the memory cells 2402 are set. Once the memorystates have been set within the memory cells 2402, the mirror clockpulse signal may be applied at step 2504. The micro-mirror 3108 willhave an established specification of the time before and after a mirrorclocking pulse that data may be loaded into the memory cell 2402.Application of the mirror clocking pulse signal will then set themirrors to their new state established by the memory at step 2506. Theprocess is completed at step 2508, and the mirror 2302 position is fixedand new data may be loaded into the memory cell 2402.

Referring now to FIG. 26, there is illustrated an intensity and phaseinterferometer for measuring the intensity and phase of the generatedbeam. One can generate spatial modes by loading computer-generatedMatlab holograms 2602 such as those described herein above andillustrated in FIGS. 31A-31H onto a DMD memory. The holograms 2602 forgenerating modes can be created by modulating a grating function with 20micro-mirrors per each period. The holograms 2602 are provided to a DMD2604. An imaging system 2606 along with an aperture 2608 separates thefirst order diffracted light into separate modes. The imaging systemincludes a laser 2610 that provides a light through a pair of lenses2612, 2614. The lens 2612 expands the light beam to lens 2614 whichcollimates the beam. A beam splitter 2616 splits the beam toward a lens2618 and mirror 2621. Lens 2618 focuses the beam through lens 2620 whichcollimates the beam through a filter 2622. The filtered beam isreflected by mirror 2624 through a second beam splitter 2626. The beamsplitter 2626 splits the beam toward a lens 2628 and a charge coupleddevice camera 2630. The charge coupled device (CCD) camera 2630 measuresthe intensity profile of the generated beam. The plane wave beamprovided to lens 2628 is focused on to the aperture 2608 to interferewith the twisted beam from the DMD. Also focused on the aperture 2608 isthe twisted beam from the DMD 2604. The beam from the DMD 2604 isprovided through a lens 2632 that also focuses on the aperture 2608. Thephase of the mode being generated is determined from the number ofspirals in the pattern and is caused by interfering the twisted beamwith a plane wave beam. Also, whether the phase is positive or negativemay be determined by whether the spirals are clockwise (positive) orcounter-clockwise (negative). A Mach-Zehnder interferometer may be usedto verify the phase pattern of the created beams. The collimated planewave provided from lens 2628 is interfered with the modes generated bythe beam from the DMD 2604 through lens 2632. This generates theinterferograms (spiral patterns) at the aperture 2608. The modesgenerated from the DMD may then be multiplexed together usingmemory-based static forks on the DLP.

Therefore, there is a possibility of using binary holograms tocoherently control both phase and amplitude of a light beam. A lownumber of pixels per each period of the binary grating results inquantization errors in encoding phase and intensity. The total number ofgrating periods with in the incident beam on the DMD 2604 sets an upperlimit on the spatial bandwidth of the generated modes. Consequently alarge number of micro-mirrors is preferable for generating high-qualitymodes. This can be achieved by using newer generations of DMDs. Anotherset of modes that are needed for OAM-based quantum key distribution isthe set of angular (ANG) modes.

Referring now to FIG. 27A, there is illustrated the manner in whichswitching between different OAM modes may be achieved in real time. Thelaser 2702 generates a collimated beam through lenses 2704 and 2706 to aDMD 2708. The DMD 2708 provides a beam that is focused by lens 2710 ontoaperture 2712. The output from the aperture 2712 is provided to a lens2714 that collimates the beam onto a mirror 2716. The collimated beam isprovided to an OAM sorter 2718 that separates the signal into variousOAM modes 2720 as detected by a computer 2722.

Referring now to FIG. 27B, there is more generally illustrated themanner in which a transmitter 2750 processes multiple data channels 2752that are passed through a cylindrical lens 2754 to a focusing lens 2756.The lens 2756 focuses the beam on a OAM sorter 2758. The collimated beamis passed through an OAM sorter 2758 for multiplexing the OAM beamstogether as multiplexed OAM beams 2764 transmission to a receiver 2764.The multiplexed OAM beams 2760 are passed through a second OAM sorter2762 at the receiver 2764 to demultiplex the beams into separate OAMchannels. The received OAM channels 2768 are passed through a lens 2766to focus the separate OAM beam channels 2768.

Using DMDs for generating OAM modes provides the ability to switchbetween different modes at very high speeds. This involves a muchsmaller number of optical elements as compared to the conventionaltechniques were OAM modes are generated using a series of separatedforked holograms and are multiplexed using beam splitters. Therefore,one can achieve dynamic switching among vortex OAM modes with differentquantum numbers. The computer-generated holograms for these modes mustbe loaded onto the memory of the DMD 2708, and the switching is achievedby using a clock signal. One can use a mode sorter to map the inputmodes to a series of separated spots. The intensity may then be measuredcorresponding to each mode using a high-bandwidth PIN detector atpositions corresponding to each mode. The DMD devices are available fora fraction of the cost of phase only spatial light modulators.

The DMD efficiency observed in a specific application depends onapplication-specific design variables such as illumination wavelength,illumination angle, projection aperture size, overfill of the DMDmicro-mirror array and so on. Overall optical efficiency of each DMD cangenerally be estimated as a product of window transmission, adiffraction efficiency, micro-mirror surface reflectivity and array fillfactor. The first three factors depend on the wavelength of theillumination source.

DLP technology uses two types of materials for DMD mirrors. The mirrormaterial for all DMD's except Type-A is Corning Eagle XG, whereas type ADMDs use Corning 7056. Both mirror types have an anti-reflectivity (AR),thin-film coating on both the top and the bottom of the window glassmaterial. AR coatings reduce reflections and increase transmissionefficiency. The DMD mirrors are designed for three transmission regions.These ranges include the ultraviolet light region from 300 nm to 400 nm,the visible light region from 400 nm to 700 nm and the near infraredlight region (NIR) from 700 nm to 2500 nm. The coating used depends onthe application. UV windows have special AR coatings designed to be moretransmissive for ultraviolet wavelengths, visible coatings for visibleDMDs and NIR coatings for NIR DMDs.

The measured data provided in the following sections reflects a typicalsingle pass transmittance through both top and bottom AR coated mirrorsurfaces with random polarization. The angle of incidence (AOI) of 0° ismeasured perpendicular to the window surface unless mentioned otherwise.With an increase in the number of window passes, the efficiency woulddecline.

FIG. 28 represents the window transmission curves for Corning 7056. Thewindow transmission response curve in this figure applies to TaipeiMDM's in their specified illumination wavelength regions. FIG. 28 showsthe UV window transmittance measured perpendicular to the window surfaceand visible window transmittance at a lie of 0° and 30°. FIGS. 29-33 arezoomed in views of the typical visible and UV AR coated windowtransmittance in their maximum transmission regions. The visible CorningEagle XG window transmission data shown in FIG. 32 applies to the DLP5500, DLP 1700, DLP 3000 and DLP 3000 DMD's. The typical transmittanceobserved in these DMD's is broadband visible region is approximately97%. The NIR Corning Eagle XG window transmission data of FIG. 33applies to the DLP 3000 NIR DMD. The typical transmittance observed inthe NIR DMD's in the broadband NIR region is approximately 96% for mostof the region with a dip toward 90% as it nears 2500 nm.

Referring now to FIG. 34, there is illustrated a configuration ofgeneration circuitry for the generation of an OAM twisted beam using ahologram within a micro-electrical mechanical device. A laser 3402generates a beam having a wavelength of approximately 543 nm. This beamis focused through a telescope 3404 and lens 3406 onto a mirror/systemof mirrors 3408. The beam is reflected from the mirrors 3408 into a DMD3410. The DMD 3410 has programmed in to its memory a one or more forkedholograms 3412 that generate a desired OAM twisted beam 3413 having anydesired information encoded into the OAM modes of the beam that isdetected by a CCD 3414. The holograms 3412 are loaded into the memory ofthe DMD 3410 and displayed as a static image. In the case of 1024×768DMD array, the images must comprise 1024 by 768 images. The controlsoftware of the DMD 3410 converts the holograms into .bmp files. Theholograms may be displayed singly or as multiple holograms displayedtogether in order to multiplex particular OAM modes onto a single beam.The manner of generating the hologram 3412 within the DMD 3410 may beimplemented in a number of fashions that provide qualitative differencesbetween the generated OAM beam 3413. Phase and amplitude information maybe encoded into a beam by modulating the position and width of a binaryamplitude grating used as a hologram. By realizing such holograms on aDMD the creation of HG modes, LG modes, OAM vortex mode or any angularmode may be realized. Furthermore, by performing switching of thegenerated modes at a very high speed, information may be encoded withinthe helicity's that are dynamically changing to provide a new type ofhelicity modulation. Spatial modes may be generated by loadingcomputer-generated holograms onto a DMD. These holograms can be createdby modulating a grating function with 20 micro mirrors per each period.

Rather than just generating an OAM beam 3413 having only a single OAMvalue included therein, multiple OAM values may be multiplexed into theOAM beam in a variety of manners as described herein below. The use ofmultiple OAM values allows for the incorporation of differentinformation into the light beam. Programmable structured light providedby the DLP allows for the projection of custom and adaptable patterns.These patterns may be programmed into the memory of the DLP and used forimparting different information through the light beam. Furthermore, ifthese patterns are clocked dynamically a modulation scheme may becreated where the information is encoded in the helicities of thestructured beams.

Referring now to FIG. 35, rather than just having the laser beam 3502shine on a single hologram multiple holograms 3504 may be generated bythe DMD 3410. FIG. 35 illustrates an implementation wherein a 4×3 arrayof holograms 3504 are generated by the DMD 3410. The holograms 3504 aresquare and each edge of a hologram lines up with an edge of an adjacenthologram to create the 4×3 array. The OAM values provided by each of theholograms 3504 are multiplexed together by shining the beam 3502 ontothe array of holograms 3504. Several configurations of the holograms3504 may be used in order to provide differing qualities of the OAM beam3413 and associated modes generated by passing a light beam through thearray of holograms 3504.

FIG. 36 illustrates various reduced binary fork holograms that may beused for applying different OAM levels to a light. FIG. 36 illustratesholograms for applying OAM light from

=1 to

=10 a period of 100.

Referring now to FIG. 37, there is illustrated the manner in which acombined use of OAM processing and polarization processing may be usedto increase the data with any particular combination of signals using aDLP system. A variety of data (Data1, Data3, Data5, Data7) 3702 havediffering OAM levels (OAM1, OAM2, OAM3, OAM4) 3704 and differing X and Ypolarizations 3706. This enables multiplexing of the signals togetherinto a polarization multiplexed OAM signal 3708. The polarizationmultiplexed OAM signal 3708 made the demultiplexed by removing the X andY polarizations 3706 and OAM to re-create the data signals 3702.

Spin angular momentum (SAM) is associated with polarization and given byσ

=±

(for circular polarization). While orbital angular momentum (OAM) isassociated with azimuthal phase of the complex electric field. Eachphoton with the azimuthal phase dependence is of the form exp(−jlØ)(1=0,±1, ±2, . . . ) and carries the OAM of l

. Therefore, with each photon we can associate a photon angular momentumdefined over computational basis states |I, σ

. Because the OAM eigenstates are mutually orthogonal, an arbitrarynumber of bits per single photon can be transmitted. The possibility togenerate/analyze states with different photon angular momentum, by usingholographic method, allows the realization of quantum states inmultidimensional Hilbert space. Because OAM states provide an infinitebasis state, while SAM states are two-dimensional only, the OAM can alsobe used to simultaneously increase the security for QKD and improve thecomputational power for quantum computing applications. We introduce thefollowing deterministic quantum qu-dit gates and modules based on photonangular momentum.

Qudit Gates

Referring now to FIG. 37B, the basic quantum modules 302 for quantumteleportation applications include the generalized-Bell-state generationmodule 304 and the QFT-module 306. The basic module for entanglementassisted QKD is either the generalized-Bell-state generation module 304or the Weyl-operator-module 312. The photon angular momentum baseduniversal quantum qudit gates, namely generalized-X, generalized-Z,generalized-CNOT qudit gates. A set of universal quantum gates is anyset of gates to which any operation possible on a quantum computer canbe reduced, that is, any other unitary operation can be expressed as afinite sequence of gates from the set. Technically this is impossiblesince the number of possible quantum gates is uncountable, whereas thenumber of finite sequences from a finite set is countable. To solve thisproblem, we only require that any quantum operation can be approximatedby a sequence of gates from this finite set. Moreover, for unitaries ona constant number of qubits, the Solovay-Kitaev theorem guarantees thatthis can be done efficiently.

Different quantum modules 302 of importance are introduced for differentapplications including (fault-tolerant) quantum computing,teleportation, QKD, and quantum error correction. Quantum Q-dit modulesinclude generalized Bell State Generation Modules 304, QFT Modules 306,Non-Binary Syndrome Calculator Modules 308, Generalized Universal Q-ditgates 310, Weyl-Operator Modules 312 and Generalized Controlled PhaseDeterministic Qudit Gate Using Optics 314, which is a key advantagecompared to probabilistic SAM based CNOT gate. Also, by describing suchgates and modules, we introduce their corresponding integrated opticsimplementation on DLP. We also introduce several entanglement assistedprotocols by using the generalized-Bell-state generation module. Theapproach is to implement all these modules in integrated optics usingmulti-dimensional qudits on DLP

Photon OAM Based Universal Qudit Gates and Quantum Modules

An arbitrary photon angular momentum state |ψ

can be represented as a linear superposition of |I, σ

-baskets as follows:

${\left. \psi \right\rangle = {\sum\limits_{I = {- L_{-}}}^{L_{+}}{\sum\limits_{\sigma = {\pm 1}}{C_{1,\sigma}\left. {I,\sigma} \right\rangle}}}},{\sum\limits_{I = {- L_{-}}}^{L_{+}}{\sum\limits_{\sigma = {\pm 1}}{C_{1,\sigma}}^{2}}}$where the |I, σ

-basekets are mutually orthogonal, namely

m,σ|n,σ

=δ _(mn)δ_(σσ′) ;m,n,∈{−L ⁻, . . . ,−1,0,1, . . . ,L _(±)};σ,σ′∈{−1,1}

Therefore, the photon angular momentum kets live in D=2(L⁻+L₊+1)dimensional Hilbert space

₂ (L⁻+L₊+1). Kets are defined as a vector in Hilbert space, especiallyas representing the state of a quantum mechanical system. Notice that inthe most general case, the number of states with negative OAM index,denoted as L does not need to be the same as the number of OAM stateswith positive OAM index. This photon angular momentum concept todescribe the photon states is different from the total angular momentumof photon defined as j

=(I+σ).

As an illustration, in total angular momentum-notation for j=4 we cannotdistinguish between |I=5, σ=−1

and |I=3, σ=1

photon angular momentum states. Therefore, the use of |I=σ

notation is more general. The SAM (circularly polarized) states can berepresented in computational base {{|H

, [V

|H

—horizontal photon, [V

-vertical photon) as follows:

${\left. {+ 1} \right\rangle = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\j\end{bmatrix}}},{\left. {+ 1} \right\rangle = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\{- j}\end{bmatrix}}}$

The SAM operator is represented by:

$S = \begin{bmatrix}0 & {- j} \\j & 0\end{bmatrix}$Clearly, right-circular (|+1

) and left-circular (|−1

states are eigenkets of this operator since:S|+1

=|+1

,S|−1

=−|−1

The OAM states (|+1

) and (|−1

) generated using, for example, the process described previously, can berepresented in reduced two-dimensional subspace, respectively, asfollows:

${\left. 1 \right\rangle = {\frac{1}{l}\begin{bmatrix}l \\0\end{bmatrix}}},\mspace{14mu}{\left. {- 1} \right\rangle = {\frac{1}{l}\begin{bmatrix}O \\{- l}\end{bmatrix}}}$

Referring now to FIG. 38, there are illustrated a number of examples ofquantum gates 3802 that generate outputs responsive to input qubits |x

applied to the input of quantum gates 3802 a, 3802 b, 3802 c and 3802 dand the input qubits |x

and |y

applied to the inputs of gate 3802 e. The outputs of the various quantumgates 3802 may be provided depending upon the type of processesimplemented within the quantum gate. While the following examples aremad with respect to qubit gates having inputs and outputs based on spinangular momentum, by applying OAM processing as described herein to theinputs and outputs the gates can operate as qudit gates operable with alarger number of input states.

In this representation shown in FIG. 38, since the mode with l=0 has noOAM value, it can be denoted as:|0

[0 0]^(T)

In this particular case, the photon angular momentum state reduces tothe SAM state only. By assuming that the OAM-ket is aligned with thedirection of propagation (z-axis), the OAM operator can be representedas:

$L_{z} = \begin{bmatrix}l & O \\O & {- l}\end{bmatrix}$

It is straightforward to verify that states |l

and |−l

are eigenkets of OAM operator L_(z):L _(z) |±I

=(±I)|±I

The spin operator S and SAM operator L_(z) satisfy the followingproperties:S ² =I ₂ ,L _(z) ² =l ² I ₂where I₂ is the identity operator. The photon angular momentum operatorcan now be defined as

$J = {{\left( {L_{z} \otimes I} \right)\;\left( {I \otimes S} \right)} = \begin{bmatrix}{IS} & O \\O & {- {IS}}\end{bmatrix}}$where the operator ⊗ denotes the tensor product. The correspondingeigenvalue equation is given by:]|±I,±1

=(±I)(±1)|±I,|±1

For convenience purpose, we can use a single indexing of photon angularmomentum, and the computational bases related to photon angular momentumstates are denoted as {|0

, |1

, . . . , |D−1

}, D=2(L⁻+L₊).

By properly selecting L⁻ and L₊ to make sure that the dimensionality Dis equal to some power of two, then the dimension D can be expressed asD=q=2^(p), where p≥1 is a prime. If the addition operation is performedover Galois field GF(2^(p)) instead “per mod D,” the set of gates thatcan be used for arbitrary operation on qudits can be defined asillustrated in FIG. 38. The F-gate corresponds to the quantum Fouriertransform (QFT) gate. Its action on ket |0

is the superposition of all basis kets with the same probabilityamplitude

${F\left. 0 \right\rangle} = {q^{- \frac{1}{2}}{\sum{\left. u \right\rangle.}}}$

Therefore, the F-gate on qudits has the same role as Hadamard gate onqu-bits. The action of generalized X- and Z-gates can be described asfollows:X(a)|x

=|x

=|x+a

,Z(b)|x

=ω ^(tr(bx)) |x

;x,a,b∈GF(q)Where x, a, b, ∈GF(q), tr(.) denotes the trace operation from GF(q) toGF(p) and ω is a p-th root of unity, namely ω=exp(j2π/p). By omittingthe SAM as a degree of freedom, since it represents a fragile source ofquantum information, the corresponding space becomes (L⁻+L₊)−dimensional(D=L⁻+L₊).

By selecting (L⁻+L₊) to be a prime P, the corresponding additionoperation represents “mod P addition,” in other words right cyclicshift. The left cyclic shift can be defined as:X(a)|x

=|x−a

At the same time, the trace operationbecomes trivial, and the action ofgeneralized Z-gate becomes

${{Z(b)}\left. m \right\rangle} = {e^{{j{(\frac{2\;\pi}{p})}}{bm}}{\left. m \right\rangle.}}$The corresponding generalized-Hadamard gate in this formalism is theF-gate since:

${F\left. n \right\rangle} = {D^{{- 1}/2}{\sum\limits_{m = 0}^{D - 1}{\omega^{{- n}\; m}\left. m \right\rangle}}}$

Therefore, all these single-qudit gates can be implemented by using theOAM as a degree of freedom. One particularly suitable technology isbased on spatial modes in few-mode fibers. As shown in FIG. 39, thebasic building blocks for that purpose are OAM multiplexers 3902, OAMde-multiplexers 3904, few-mode fiber itself, and a series ofelectro-optical modulators 3906. Since few-mode fibers are notcompatible with integrated optics, single-qudit OAM-based gate can bemodified as described below.

By using the basic qudit gates shown in FIG. 38, a more complicatedquantum modules can be implemented as shown in FIG. 40. FIG. 40illustrates an OAM-based qudit teleportation module 4002. The module4002 comprises an OAM based quantum qudits teleportation module 4004that includes a generalized bell state generator module 4006. Thegeneralized bell state generator module 4006 comprises a generalizedF-gate 4008 and a generalized X-gate 4010. Input 4012 is applied to thegeneralized F-gate 4008 and input 4014 is provided to generalized X-gate4010. An output of the generalized F-gate 4008 is also provided as asecond input to the generalized X-gate 4010. The OAM based quantum quditteleportation module 4004 in addition to including the generalized bellstate generator module 4006 provides the output from each of the F-gates4008 to a generalized Xnot-gate 4012 and the output from generalizedX-gate 4010 to Xnot-gate 4014. A second input to generalized Xnot-gate4012 is provided from input 4016. Input 4016 also is provided togeneralized F-gate 4018. Xnot-gate 4012 also provides an input tomeasurement circuitry 4024 for preforming a measurement on the output ofa gate. The output of generalized F-gate 4018 is provided as an input tomeasurement circuitry 4020 and as an input to generalized Z-gate 4022that also has an input therein from Xnot-gate 4014. The output of thegeneralized Z-gate 4022 provides the output 4026 of the OAM-wastequantum qudit teleportation

module 4002. |B₀₀

Can be used to denote simplest generalized Bell baseket as follows:

$\left. B_{00} \right\rangle = {{{{CNOT}\left( {F \otimes I} \right)}\left. 0 \right\rangle\left. 0 \right\rangle} = {D^{- \frac{1}{2}}{\sum\limits_{m = 0}^{D - 1}{\left. m \right\rangle\left. m \right\rangle}}}}$

Another example as shown in FIG. 41 illustrates the syndrome calculatormodule 4102. The syndrome calculator module 4102 is important infault-tolerant computing and quantum error correction. The syndromecalculator module 4102 includes an |x

input 4104 and an |y

input 4106. The |y

input 4106 is provided to a generalized F-gate 4108. The output of thegeneralzed F-gate 4108 is provided to an inverse G-function gate 4110having its output connected to adder circuit 4112 that adds the outputsof |x

input with the output of gate 4110. The adder circuit 4112 also receivesan input from the |x

input 4104 and has its output connected to a generalized G-gate 4114.The output of the generalized G-gate is connected to the input of ageneralized F-gate 4116. The output of the generalized F-gate 4116 isconnected to the input of an inverse G-function gate 4118 whose outputis connected to an adder circuit 4120 that adds the outputs of |x

input with the output of gate 4118. The adder circuit 4120 is alsoconnect did to the |x

input 4104. The output of the adder circuit 4120 is connected to theinput of and the generalized G-gate 4122 whose output is provided as anout node 4124. The output of the |x

input 4104 is also provided as an output 4126.

In both applications, the syndrome can be determined to identify thequantum error based on syndrome measurements in accordance with thescheme illustrated by the module 4202 illustrated in FIG. 42. An |0

input 4204 is provided to the input of a generalized F-gate 4206. Theoutput of the generalized F-gate 4206 is provided to the input ofanother generalized F-gate 4208 into the input of a syndrome calculatormodule 4210 function module 4210 provides a pair of outputs to a furtherfunction module 4212 function module 4212 also receives an |Xn

input 4214. The output of generalized F-gate 4208 provided tomeasurement circuitry 4214.

In this syndrome decoding module 4202, the syndrome calculator module4210 S(b_(i), a_(i)) corresponds to the i-th generatorg_(i)=[a_(i)|b_(i)] of a quantum-check matrix of a nonbinary quantumerror correction code:

${A = \begin{pmatrix}(A)_{1} \\\vdots \\(A)_{N - K}\end{pmatrix}},{{(A)_{1} = {g_{i} = {\left( {{ai}❘b_{i}} \right) = {\left\lbrack {O\mspace{14mu}\ldots\mspace{14mu} O\; a_{i}\mspace{14mu}\ldots\mspace{14mu} b_{N}} \right\rbrack \in {{GF}(q)}^{2N}}}}};a_{i}},b_{i},{\in {{GF}(q)}}$In the above equation, the parameter a_(i) is used to denote the actionof X(a_(i)) qudit gate on the i-th qudit location, while with b_(i) theaction of Z(b_(i)) qudit gate on the i-th qudit location is denoted.Arbitrary error belongs to the multiplicative Pauli group (error group)on qudits G_(N)={ω^(c)X(a)Z(b)|a, b ∈GF(q)^(N)}. By representing theerror operator as e=(c|d), corresponding to E=ω^(c)X(c)Z(d), thesyndrome can be calculated as S(E)=S(c, d)=eA^(T).

The quantum circuit of FIG. 42 will provide non-zero measurement if adetectible error does not commute with a multiple of g_(i). Thecorrectable qudit error is mapping the code space to qK-dimensionalsubspace of qN-dimensional Hilbert space. Since there are N−Kgenerators, or equivalently syndrome positions, there are q^(N-K)different cosets. All qudit errors belonging to the same coset have thesame syndrome. By selecting the most probable qudit error for the cosetrepresentative, which is typically the lowest weight error, The quditerror can be uniquely identified and consequently perform the errorcorrection action. Alternatively, the maximum-likelihood decoding can beused. However, the decoding complexity would be significantly higher.

The implementation of OAM-based single-qudit and generalized-CNOT gateswill now be more fully described. The OAM-based single-qudit gate can beimplemented in integrated optics, with the help of computer generatedholograms implemented using DLP as described above. The quantum staterepresented by |ψ

=Σ_(I)′|I

arrives at the input of a single-qudit gate. Referring now to FIG. 43,there is illustrated a block diagram of a quantum computer implementedaccording to the present disclosure. The quantum computer 4300 receivesinput data 4302 and provides output data 4304 responsive thereto. Thequantum computer utilizes a number of components in order to providethis processing functionality. The quantum computer 4300 includes acombination of quantum gates 4306 and quantum modules 4308 for carryingout the processing functionalities. The structural nature of the quantumgates 4306 and quantum modules 4308 are as described herein. Signalstransmitted within the quantum computer 4300 may make a use of acombination of orbital angular momentum processing 130 and spin angularmomentum processing 4302. The particular nature of the orbital angularmomentum processing 4310 and spin angular momentum processing 4312 areas described herein. The OAM processing of signals is assisted utilizinga DLP or other light processing system 4314 as described herein below.

As shown previously in FIG. 39, in OAM demultiplexers 3904, the baseketsare separated, before processing, by a set of electro-optical modulators3606 (E/O MODs). The required phase shift and/or amplitude change areintroduced by the electro-optical modulators 3606 to perform the desiredsingle-qudit operation. The basekets are recombined into single-qudit inthe OAM multiplexer 3902 to obtain the output quantum state |ψ

=Σ_(I=0) ^(D-1)C_(I)′|I

. As an illustration, the F-gate is obtained by implementing the FEOmodulator 3606 as a concatenation of an attenuator to introduceattenuation D^(−1/2) and a phase modulator 3906 to introduce the phaseshift −(2π/p)nm in the m-th branch. The generalized Z(b) gate isobtained by introducing the phase shift (2π/p)bm in the m-th branch bycorresponding phase modulator. The generalized X(a) gate is obtained byusing the CGH (Computer Generated Hologram) in the m-th brunch tointroduce azimuthal phase shift of the form exp (jaϕ).

Referring now to FIG. 44, by implementing an E/O modulator 3906 as aconcatenation of an amplitude modulator 4402 and a phase modulator 4404or a single IQ modulator, the single-qudit gate is straightforward toinitialize to arbitrary state by properly adjusting amplitude and phasechanges in each branch. For instance, the superposition of all basiskets with the same probability amplitude is obtained by simplyintroducing the amplitude change D^(−1/2) in each branch, while settingthe phase shift to zero.

Referring now to FIG. 45, the generalized-CNOT gate 4502 in which thepolarization qubit 4404 serves as a control qubit and OAM qudit 4406 asthe target qudit is a quantum gate that is an essential component in theconstruction of a quantum computer. A generalized CNOT gate can be usedto entangle or disentangle EPR states. Any quantum circuit can besimulated to an arbitrary degree of accuracy using a combination of CNOTand single qudit rotations. Here we are concerned instead withimplementation in which both control 4404 and target 4406 qudit are OAMstates. The holographic interaction between OAM |I_(C) and OAM |I_(T)states can be described by the following Hamiltonian:H=gJ _(C) J _(T)

Referring now to FIG. 46, there is illustrated the operations of theCNOT gate on a quantum register consisting of two qubits. The CNOT gateflips the second qubit (the target qubit) 4406 if and only if the firstqubit (the control qubit) 4404 is |1

.

The corresponding time-evolution operator is given by:U(t)=exp[−jtgJ_(C) J _(T)]=exp[−jtgI_(C) I _(T) I ₄]By choosing gt=2π/(L⁻+L₊), the following unitary operator is obtained:

${C_{I_{C}}\left( {Z\left( I_{T} \right)} \right)} = {\exp\left\lbrack {{- j}\frac{2\;\pi}{L_{-} + L_{+}}I_{C}I_{T}I_{4}} \right\rbrack}$which is clearly generalized-controlled-Z operator. A ceneralized-CNOToperator can be obtained by transforming the generalized-controlled-Zoperator as follows:CNOT_(I) _(C) _(,I) _(T) =C _(I) _(C) (X(I _(T)))=(I⊗F ^(†))C _(I) _(C)(Z(I _(T)))(I⊗F)

Since this OAM interaction does not require the use of nonlinearcrystals or highly nonlinear fibers, the OAM states represent aninteresting qudit representation for quantum computing, quantumteleportation, and QKD applications. Given that generalized-X,generalized-Z, and generalized-CNOT gates represent the set of universalquantum gates, arbitrary quantum computation is possible by employingthe OAM gates described. It has been shown that the following qudit setof gates including generalized-X, generalized-Z, and eithergeneralized-CNOT or generalized controlled-phase is universal.

Bell State

It is also possible to generate the generalized Bell state |B₀₀

. An arbitrary generalized-Bell state is generated in the followingmanner. By applying the same gates as in the figures, but now on kets |n

and |m

, to obtain:

${\left( {F \otimes I} \right)\left. n \right\rangle\left. m \right\rangle} = {D^{- \frac{1}{2}}{\sum\limits_{m = 0}^{D - 1}{\omega^{{- n}k}\left. k \right\rangle\left. m \right\rangle}}}$

By applying now the generalized-CNOT gate, the desired generalized—Bellstate |B_(mn)

is obtamned:

${{CNOT}\mspace{11mu}\left( {F \otimes I} \right)\left. n \right\rangle\left. m \right\rangle} = {D^{- \frac{1}{2}}{\underset{m = 0}{\sum\limits^{D - 1}}{\omega^{- {nk}}\left. k \right\rangle\mspace{11mu}\left. {m + k} \right\rangle}}}$

Another approach to generate |B_(mn)

is to start with |B₀₀

and apply the Weyl-gate, defined as W_(mn)=Σ_(d=0) ^(D-1)ω^(−dn)|d+m

d|, on second qudit in entangled pair.

${\left( {I \otimes W_{mn}} \right)\left. B_{00} \right\rangle} = {{D^{- \frac{1}{2}}{\sum\limits_{k = 0}^{D - 1}{\left. k \right\rangle\left( {\sum\limits_{d = 0}^{D - 1}{\omega^{- {dn}}\left. {d = m} \right\rangle\left\langle d \right.}} \right)\left\langle k \right.}}} = {D^{- \frac{1}{2}}{\sum\limits_{k = 0}^{D - 1}{\omega^{- {kn}}\left. k \right\rangle\left. {k + m} \right\rangle}}}}$The Weyl-gate can easily implemented by moving from d-th to (d+m) mod Dbranch and introducing the phase shift −2πdk/D in that branch.

Now, all elements required to formulate entanglement assisted protocolsbased on OAM are available. The multidimensional QKD is described morefully below. The proposed qudit gates and modules can be employed toimplement important quantum algorithms more efficiently. For example,the basic module to implement the Grover search algorithm, performing asearch for an entry in unstructured database, is the Grover quditoperator, which can be represented as:G=(2F ^(⊗n) |O

O|F ^(⊗n) −I)Owhere F is the QFT qudit gate and O is the oracle operator, defined as:O|x

=(−1)^(ƒ(x)) |x

;x=(x ₁ x ₂ . . . x _(N)),x _(i) ∈GF(q)with ƒ(x) being the search function, generating 1 when the searched itemis found, and zero otherwise. Shor factorization and Simon's algorithmsare also straightforward to generalize.

The main problem related to OAM-based gates is the imperfect generationof OAM modes (especially using DLP). Currently existing CGHs stillexhibit the measurable OAM crosstalk. On the other hand, the OAM is verystable degree of freedom, which does not change much unless the OAMmodes are propagated over the atmospheric turbulence channels. Moreover,OAM states are preserved after kilometers-length-scale propagation inproperly designed optical fibers. The noise affects the photons carryingOAM in the same fashion as the polarization states of photons areaffected. The noise is more relevant in photon-number-sates-basedoptical quantum computing than in OAM-based quantum computing.

Quantum Key Distribution

As described above, one manner for using OAM based quantum computinginvolves the use in processes such as Quantum Key Distribution (QKD). Incurrent QKD system, the systems are very slow. By implementing the abovesystem of quantum gate computing using OAM, system can increase securityand throughput communications while increasing the capacity of computingand processing of the system. The QKD operations would be implemented ina Quantum Module implementing the processes as described above.Referring now to FIG. 47, there is illustrated a further improvement ofa system utilizing orbital angular momentum processing, LaguerreGaussian processing, Hermite Gaussian processing or processing using anyorthogonal functions. In the illustration of FIG. 47, a transmitter 4702and receiver 4704 are interconnected over an optical link 4706. Theoptical link 4706 may comprise a fiber-optic link or a free-space opticlink as described herein above. The transmitter receives a data stream4708 that is processed via orbital angular momentum processing circuitry4710. The orbital angular momentum processing circuitry 4710 provideorbital angular momentum twist to various signals on separate channelsas described herein above. In some embodiments, the orbital angularmomentum processing circuitry may further provide multi-layer overlaymodulation to the signal channels in order to further increase systembandwidth.

The OAM processed signals are provided to quantum key distributionprocessing circuitry 4712. The quantum key distribution processingcircuitry 4712 utilizes the principals of quantum key distribution aswill be more fully described herein below to enable encryption of thesignal being transmitted over the optical link 4706 to the receiver4704. The received signals are processed within the receiver 4704 usingthe quantum key distribution processing circuitry 4714. The quantum keydistribution processing circuitry 4714 decrypts the received signalsusing the quantum key distribution processing as will be more fullydescribed herein below. The decrypted signals are provided to orbitalangular momentum processing circuitry 4716 which removes any orbitalangular momentum twist from the signals to generate the plurality ofoutput signals 4718. As mentioned previously, the orbital angularmomentum processing circuitry 4716 may also demodulate the signals usingmultilayer overlay modulation included within the received signals.

Orbital angular momentum in combination with optical polarization isexploited within the circuit of FIG. 47 in order to encode informationin rotation invariant photonic states, so as to guarantee fullindependence of the communication from the local reference frames of thetransmitting unit 4702 and the receiving unit 4704. There are variousways to implement quantum key distribution (QKD), a protocol thatexploits the features of quantum mechanics to guarantee unconditionalsecurity in cryptographic communications with error rate performancesthat are fully compatible with real world application environments.

Encrypted communication requires the exchange of keys in a protectedmanner. This key exchanged is often done through a trusted authority.Quantum key distribution is an alternative solution to the keyestablishment problem. In contrast to, for example, public keycryptography, quantum key distribution has been proven to beunconditionally secure, i.e., secure against any attack, even in thefuture, irrespective of the computing power or in any other resourcesthat may be used. Quantum key distribution security relies on the lawsof quantum mechanics, and more specifically on the fact that it isimpossible to gain information about non-orthogonal quantum stateswithout perturbing these states. This property can be used to establishrandom keys between a transmitter and receiver, and guarantee that thekey is perfectly secret from any third party eavesdropping on the line.

In parallel to the “full quantum proofs” mentioned above, the securityof QKD systems has been put on stable information theoretic footing,thanks to the work on secret key agreements done in the framework ofinformation theoretic cryptography and to its extensions, triggered bythe new possibilities offered by quantum information. Referring now toFIG. 48, within a basic QKD system, a QKD link 4802 is a point to pointconnection between a transmitter 4804 and a receiver 4806 that want toshare secret keys. The QKD link 4802 is constituted by the combinationof a quantum channel 4808 and a classic channel 4810. The transmitter4804 generates a random stream of classical bits and encodes them into asequence of non-orthogonal states of light that are transmitted over thequantum channel 4808. Upon reception of these quantum states, thereceiver 4806 performs some appropriate measurements leading thereceiver to share some classical data over the classical link 4810correlated with the transmitter bit stream. The classical channel 4810is used to test these correlations.

If the correlations are high enough, this statistically implies that nosignificant eavesdropping has occurred on the quantum channel 4808 andthus, that has a very high probability, a perfectly secure, symmetrickey can be distilled from the correlated data shared by the transmitter4804 and the receiver 4806. In the opposite case, the key generationprocess has to be aborted and started again. The quantum keydistribution is a symmetric key distribution technique. Quantum keydistribution requires, for authentication purposes, that the transmitter4804 and receiver 4806 share in advance a short key whose length scalesonly logarithmically in the length of the secret key generated by an OKDsession.

Quantum key distribution on a regional scale has already beendemonstrated in a number of countries. However, free-space optical linksare required for long distance communication among areas which are notsuitable for fiber installation or for moving terminals, including theimportant case of satellite based links. The present approach exploitsspatial transverse modes of the optical beam, in particular of the OAMdegree of freedom, in order to acquire a significant technical advantagethat is the insensitivity of the communication to relevant alignment ofthe user's reference frames. This advantage may be very relevant forquantum key distribution implementation to be upgraded from the regionalscale to a national or continental one, or for links crossing hostileground, and even for envisioning a quantum key distribution on a globalscale by exploiting orbiting terminals on a network of satellites.

The OAM Eigen modes are characterized by a twisted wavefront composed of“l” intertwined helices, where “l” is an integer, and by photonscarrying “+l

” of (orbital) angular

momentum, in addition to the more usual spin angular momentum (SAM)associated with polarization. The potentially unlimited value of “l”opens the possibility to exploit OAM also for increasing the capacity ofcommunication systems (although at the expense of increasing also thechannel cross-section size), and terabit classical data transmissionbased on OAM multiplexing can be demonstrated both in free-space andoptical fibers. Such a feature can also be exploited in the quantumdomain, for example to expand the number of qubits per photon, or toachieve new functions, such as the rotational invariance of the qubits.

In a free-space QKD, two users (Alice and Bob) must establish a sharedreference frame (SRF) in order to communicate with good fidelity. Indeedthe lack of a SRF is equivalent to an unknown relative rotation whichintroduces noise into the quantum channel, disrupting the communication.When the information is encoded in photon polarization, such a referenceframe can be defined by the orientations of Alice's and Bob's“horizontal” linear polarization directions. The alignment of thesedirections needs extra resources and can impose serious obstacles inlong distance free space QKD and/or when the misalignment varies intime. As indicated, we can solve this by using rotation invariantstates, which remove altogether the need for establishing a SRF. Suchstates are obtained as a particular combination of OAM and polarizationmodes (hybrid states), for which the transformation induced by themisalignment on polarization is exactly balanced by the effect of thesame misalignment on spatial modes. These states exhibit a globalsymmetry under rotations of the beam around its axis and can bevisualized as space-variant polarization states, generalizing thewell-known azimuthal and radial vector beams, and forming atwo-dimensional Hilbert space. Moreover, this rotation-invariant hybridspace can be also regarded as a decoherence-free subspace of thefour-dimensional OAM-polarization product Hilbert space, insensitive tothe noise associated with random rotations.

The hybrid states can be generated by a particular space-variantbirefringent plate having topological charge “q” at its center, named“q-plate”. In particular, a polarized Gaussian beam (having zero OAM)passing through a q-plate with q=½ will undergo the followingtransformation:(α|R

+β|R

)_(π)⊗|0

_(O) →α|L

| _(π) ⊗|r

_(O) +β|R

_(π) ⊗|l

_(O)

|L>_(πx−) and |R>_(π) denote the left and right circular polarizationstates (eigenstates of SAM with eigenvalues “±

”), |0>_(O) represents the transverse Gaussian mode with zero OAM andthe |L>_(O−) and |R>_(O) eigenstates of OAM with |l|=1 and witheigenvalues “±l

”). The states appearing on the right hand side of equation arerotation-invariant states. The reverse operation to this can be realizedby a second q-plate with the same q. In practice, the q-plate operatesas an interface between the polarization space and the hybrid one,converting qubits from one space to the other and vice versa in auniversal (qubit invariant) way. This in turn means that the initialencoding and final decoding of information in our QKD implementationprotocol can be conveniently performed in the polarization space, whilethe transmission is done in the rotation-invariant hybrid space.

OAM is a conserved quantity for light propagation in vacuum, which isobviously important for communication applications. However, OAM is alsohighly sensitive to atmospheric turbulence, a feature which limits itspotential usefulness in many practical cases unless new techniques aredeveloped to deal with such issues.

Quantum cryptography describes the use of quantum mechanical effects (inparticular quantum communication and quantum computation) to performcryptographic tasks or to break cryptographic systems. Well-knownexamples of quantum cryptography are the use of quantum communication toexchange a key securely (quantum key distribution) and the hypotheticaluse of quantum computers that would allow the breaking of variouspopular public-key encryption and signature schemes (e.g., RSA).

The advantage of quantum cryptography lies in the fact that it allowsthe completion of various cryptographic tasks that are proven to beimpossible using only classical (i.e. non-quantum) communication. Forexample, quantum mechanics guarantees that measuring quantum datadisturbs that data; this can be used to detect eavesdropping in quantumkey distribution.

Quantum key distribution (QKD) uses quantum mechanics to guaranteesecure communication. It enables two parties to produce a shared randomsecret key known only to them, which can then be used to encrypt anddecrypt messages.

An important and unique property of quantum distribution is the abilityof the two communicating users to detect the presence of any third partytrying to gain knowledge of the key. This results from a fundamentalaspect of quantum mechanics: the process of measuring a quantum systemin general disturbs the system. A third party trying to eavesdrop on thekey must in some way measure it, thus introducing detectable anomalies.By using quantum superposition or quantum entanglement and transmittinginformation in quantum states, a communication system can be implementedwhich detects eavesdropping. If the level of eavesdropping is below acertain threshold, a key can be produced that is guaranteed to be secure(i.e. the eavesdropper has no information about it), otherwise no securekey is possible and communication is aborted.

The security of quantum key distribution relies on the foundations ofquantum mechanics, in contrast to traditional key distribution protocolwhich relies on the computational difficulty of certain mathematicalfunctions, and cannot provide any indication of eavesdropping orguarantee of key security.

Quantum key distribution is only used to reduce and distribute a key,not to transmit any message data. This key can then be used with anychosen encryption algorithm to encrypt (and decrypt) a message, which istransmitted over a standard communications channel. The algorithm mostcommonly associated with QKD is the one-time pad, as it is provablysecure when used with a secret, random key.

Quantum communication involves encoding information in quantum states,or qubits, as opposed to classical communication's use of bits. Usually,photons are used for these quantum states and thus are applicable withinquantum computing systems. Quantum key distribution exploits certainproperties of these quantum states to ensure its security. There areseveral approaches to quantum key distribution, but they can be dividedinto two main categories, depending on which property they exploit. Thefirst of these are prepare and measure protocol. In contrast toclassical physics, the act of measurement is an integral part of quantummechanics. In general, measuring an unknown quantum state changes thatstate in some way. This is known as quantum indeterminacy, and underliesresults such as the Heisenberg uncertainty principle, informationdistribution theorem, and no cloning theorem. This can be exploited inorder to detect any eavesdropping on communication (which necessarilyinvolves measurement) and, more importantly, to calculate the amount ofinformation that has been intercepted. Thus, by detecting the changewithin the signal, the amount of eavesdropping or information that hasbeen intercepted may be determined by the receiving party.

The second category involves the use of entanglement based protocols.The quantum states of two or more separate objects can become linkedtogether in such a way that they must be described by a combined quantumstate, not as individual objects. This is known as entanglement, andmeans that, for example, performing a measurement on one object affectsthe other object. If an entanglement pair of objects is shared betweentwo parties, anyone intercepting either object alters the overallsystem, revealing the presence of a third party (and the amount ofinformation that they have gained). Thus, again, undesired reception ofinformation may be determined by change in the entangled pair of objectsthat is shared between the parties when intercepted by an unauthorizedthird party.

One example of a quantum key distribution (QKD) protocol is the BB84protocol. The BB84 protocol was originally described using photonpolarization states to transmit information. However, any two pairs ofconjugate states can be used for the protocol, and optical fiber-basedimplementations described as BB84 can use phase-encoded states. Thetransmitter (traditionally referred to as Alice) and the receiver(traditionally referred to as Bob) are connected by a quantumcommunication channel which allows quantum states to be transmitted. Inthe case of photons, this channel is generally either an optical fiber,or simply free-space, as described previously with respect to FIG. 47.In addition, the transmitter and receiver communicate via a publicclassical channel, for example using broadcast radio or the Internet.Neither of these channels needs to be secure. The protocol is designedwith the assumption that an eavesdropper (referred to as Eve) caninterfere in any way with both the transmitter and receiver.

Referring now to FIG. 49, the security of the protocol comes fromencoding the information in non-orthogonal states. Quantum indeterminacymeans that these states cannot generally be measured without disturbingthe original state. BB84 uses two pair of states 4902, each pairconjugate to the other pair to form a conjugate pair 4904. The twostates 4902 within a pair 4904 are orthogonal to each other. Pairs oforthogonal states are referred to as a basis. The usual polarizationstate pairs used are either the rectilinear basis of vertical (0degrees) and horizontal (90 degrees), the diagonal basis of 45 degreesand 135 degrees, or the circular basis of left handedness and/or righthandedness. Any two of these basis are conjugate to each other, and soany two can be used in the protocol. In the example of FIG. 50,rectilinear basis are used at 5002 and 5004, respectively, and diagonalbasis are used at 5006 and 5008.

The first step in BB84 protocol is quantum transmission. Referring nowto FIG. 51 wherein there is illustrated a flow diagram describing theprocess, wherein the transmitter creates a random bit (0 or 1) at step5102, and randomly selects at 5104 one of the two basis, eitherrectilinear or diagonal, to transmit the random bit. The transmitterprepares at step 5106 a photon polarization state depending both on thebit value and the selected basis. So, for example, a 0 is encoded in therectilinear basis (+) as a vertical polarization state and a 1 isencoded in a diagonal basis (X) as a 135 degree state. The transmittertransmits at step 5108 a single proton in the state specified to thereceiver using the quantum channel. This process is repeated from therandom bit stage at step 5102 with the transmitter recording the state,basis, and time of each photon that is sent over the optical link.

According to quantum mechanics, no possible measurement distinguishesbetween the four different polarization states 5002 through 5008 of FIG.50, as they are not all orthogonal. The only possible measurement isbetween any two orthogonal states (and orthonormal basis). So, forexample, measuring in the rectilinear basis gives a result of horizontalor vertical. If the photo was created as horizontal or vertical (as arectilinear eigenstate), then this measures the correct state, but if itwas created as 45 degrees or 135 degrees (diagonal eigenstate), therectilinear measurement instead returns either horizontal or vertical atrandom. Furthermore, after this measurement, the proton is polarized inthe state it was measured in (horizontal or vertical), with all of theinformation about its initial polarization lost.

Referring now to FIG. 52, as the receiver does not know the basis thephotons were encoded in, the receiver can only select a basis at randomto measure in, either rectilinear or diagonal. At step 5202, thetransmitter does this for each received photon, recording the timemeasurement basis used and measurement result at step 5204. At step5206, a determination is made if there are further protons present and,if so, control passes back to step 5202. Once inquiry step 5206determines the receiver had measured all of the protons, the transceivercommunicates at step 5208 with the transmitter over the publiccommunications channel. The transmitter broadcast the basis for eachphoton that was sent at step 5210 and the receiver broadcasts the basiseach photon was measured in at step 5212. Each of the transmitter andreceiver discard photon measurements where the receiver used a differentbasis at step 5214 which, on average, is one-half, leaving half of thebits as a shared key, at step 5216. This process is more fullyillustrated in FIG. 53.

The transmitter transmits the random bit 01101001. For each of thesebits respectively, the transmitter selects the sending basis ofrectilinear, rectilinear, diagonal, rectilinear, diagonal, diagonal,diagonal, and rectilinear. Thus, based upon the associated random bitsselected and the random sending basis associated with the signal, thepolarization indicated in line 5202 is provided. Upon receiving thephoton, the receiver selects the random measuring basis as indicated inline 5304. The photon polarization measurements from these basis willthen be as indicated in line 5306. A public discussion of thetransmitted basis and the measurement basis are discussed at 5308 andthe secret key is determined to be 0101 at 5310 based upon the matchingbases for transmitted photons 1, 3, 6, and 8.

Referring now to FIG. 54, there is illustrated the process fordetermining whether to keep or abort the determined key based uponerrors detected within the determined bit string. To check for thepresence of eavesdropping, the transmitter and receiver compare acertain subset of their remaining bit strings at step 5402. If a thirdparty has gained any information about the photon's polarization, thisintroduces errors within the receiver's measurements. If more than Pbits differ at inquiry step 5404, the key is aborted at step 5406, andthe transmitter and receiver try again, possibly with a differentquantum channel, as the security of the key cannot be guaranteed. P ischosen so that if the number of bits that is known to the eavesdropperis less than this, privacy amplification can be used to reduce theeavesdropper's knowledge of the key to an arbitrarily small amount byreducing the length of the key. If inquiry step 5404 determines that thenumber of bits is not greater than P, then the key may be used at step5408.

The E91 protocol comprises another quantum key distribution scheme thatuses entangled pairs of protons. This protocol may also be used withentangled pairs of protons using orbital angular momentum processing,Laguerre Gaussian processing, Hermite Gaussian processing or processingusing any orthogonal functions for Q-bits. The entangled pairs can becreated by the transmitter, by the receiver, or by some other sourceseparate from both of the transmitter and receiver, including aneavesdropper. The photons are distributed so that the transmitter andreceiver each end up with one photon from each pair. The scheme relieson two properties of entanglement. First, the entangled states areperfectly correlated in the sense that if the transmitter and receiverboth measure whether their particles have vertical or horizontalpolarizations, they always get the same answer with 100 percentprobability. The same is true if they both measure any other pair ofcomplementary (orthogonal) polarizations. However, the particularresults are not completely random. It is impossible for the transmitterto predict if the transmitter, and thus the receiver, will get verticalpolarizations or horizontal polarizations. Second, any attempt ateavesdropping by a third party destroys these correlations in a way thatthe transmitter and receiver can detect. The original Ekert protocol(E91) consists of three possible states and testing Bell inequalityviolation for detecting eavesdropping.

Presently, the highest bit rate systems currently using quantum keydistribution demonstrate the exchange of secure keys at 1 Megabit persecond over a 20 kilometer optical fiber and 10 Kilobits per second overa 100 kilometer fiber.

The longest distance over which quantum key distribution has beendemonstrated using optical fiber is 148 kilometers. The distance is longenough for almost all of the spans found in today's fiber-opticnetworks. The distance record for free-space quantum key distribution is134 kilometers using BB84 enhanced with decoy states.

Referring now to FIG. 55, there is illustrated a functional blockdiagram of a transmitter 5502 and receiver 5504 that can implementalignment of free-space quantum key distribution. The system canimplement the BB84 protocol with decoy states. The controller 5506enables the bits to be encoded in two mutually unbiased bases Z={|0>,|1>} and X={|+>, |−>}, where |0> and |1> are two orthogonal statesspanning the qubit space and |±

=1/√2 (|0

±|1

). The transmitter controller 5506 randomly chooses between the Z and Xbasis to send the classical bits 0 and 1. Within hybrid encoding, the Zbasis corresponds to {|L

_(π)⊗|r

_(O), |R

_(π)⊗|l

_(O)} while the X basis states correspond to 1/√2 (|L

_(π)⊗|r

_(O)±|R

_(π)⊗|l

_(O)). The transmitter 5502 uses four different polarized attenuatedlasers 5508 to generate quantum bits through the quantum bit generator5510. Photons from the quantum bit generator 5510 are delivered via asingle mode fiber 5512 to a telescope 5514. Polarization states |H>,|V>, |R>, |L> are transformed into rotation invariant hybrid states bymeans of a q-plate 5516 with q=½. The photons can then be transmitted tothe receiving station 5504 where a second q-plate transform 5518transforms the signals back into the original polarization states |H>,|V>, |R>, |L>, as defined by the receiver reference frame. Qubits canthen be analyzed by polarizers 5520 and single photon detectors 5522.The information from the polarizers 5520 and photo detectors 5522 maythen be provided to the receiver controller 5524 such that the shiftedkeys can be obtained by keeping only the bits corresponding to the samebasis on the transmitter and receiver side as determined bycommunications over a classic channel between the transceivers 5526,5528 in the transmitter 5502 and receiver 5504.

Referring now to FIG. 56, there is illustrated a network cloud basedquantum key distribution system including a central server 5602 andvarious attached nodes 5604 in a hub and spoke configuration. Trends innetworking are presenting new security concerns that are challenging tomeet with conventional cryptography, owing to constrained computationalresources or the difficulty of providing suitable key management. Inprinciple, quantum cryptography, with its forward security andlightweight computational footprint, could meet these challenges,provided it could evolve from the current point to point architecture toa form compatible with multimode network architecture. Trusted quantumkey distribution networks based on a mesh of pointtopoint links lacksscalability, require dedicated optical fibers, are expensive and notamenable to mass production since they only provide one of thecryptographic functions, namely key distribution needed for securecommunications. Thus, they have limited practical interest.

A new, scalable approach such as that illustrated in FIG. 56 providesquantum information assurance that is network based quantumcommunications which can solve new network security challenges. In thisapproach, a BB84 type quantum communication between each of N clientnodes 5604 and a central sever 5602 at the physical layer support aquantum key management layer, which in turn enables secure communicationfunctions (confidentiality, authentication, and nonrepudiation) at theapplication layer between approximately N2 client pairs. This networkbased communication “hub and spoke” topology can be implemented in anetwork setting, and permits a hierarchical trust architecture thatallows the server 5602 to act as a trusted authority in cryptographicprotocols for quantum authenticated key establishment. This avoids thepoor scaling of previous approaches that required a pre-existing trustrelationship between every pair of nodes. By making a server 5602, asingle multiplex QC (quantum communications) receiver and the clientnodes 5604 QC transmitters, this network can simplify complexity acrossmultiple network nodes. In this way, the network based quantum keydistribution architecture is scalable in terms of both quantum physicalresources and trust. One can at time multiplex the server 5602 withthree transmitters 5604 over a single mode fiber, larger number ofclients could be accommodated with a combination of temporal andwavelength multiplexing as well as orbital angular momentum multiplexedwith wave division multiplexing to support much higher clients.

Referring now to FIGS. 57 and 58, there are illustrated variouscomponents of multi-user orbital angular momentum based quantum keydistribution multi-access network. FIG. 57 illustrates a high speedsingle photon detector 5702 positioned at a network node that can beshared between multiple users 5704 using conventional networkarchitectures, thereby significantly reducing the hardware requirementsfor each user added to the network. In an embodiment, the single photondetector 5702 may share up to 64 users. This shared receiverarchitecture removes one of the main obstacles restricting thewidespread application of quantum key distribution. The embodimentpresents a viable method for realizing multi-user quantum keydistribution networks with resource efficiency.

Referring now also to FIG. 58, in a nodal quantum key distributionnetwork, multiple trusted repeaters 5802 are connected via point topoint links 5804 between node 5806. The repeaters are connected viapoint to point links between a quantum transmitter and a quantumreceiver. These point to point links 5804 can be realized using longdistance optical fiber lengths and may even utilize ground to satellitequantum key distribution communication. While point to point connections5804 are suitable to form a backbone quantum core network, they are lesssuitable to provide the last-mile service needed to give a multitude ofusers access to the quantum key distribution infrastructure.Reconfigurable optical networks based on optical switches or wavelengthdivision multiplexing may achieve more flexible network structures,however, they also require the installation of a full quantum keydistribution system per user which is prohibitively expensive for manyapplications.

The quantum key signals used in quantum key distribution need onlytravel in one direction along a fiber to establish a secure key betweenthe transmitter and the receiver. Single photon quantum key distributionwith the sender positioned at the network node 5806 and the receiver atthe user premises therefore lends itself to a passive multi-user networkapproach. However, this downstream implementation has two majorshortcomings. Firstly, every user in the network requires a singlephoton detector, which is often expensive and difficult to operate.Additionally, it is not possible to deterministically address a user.All detectors, therefore, have to operate at the same speed as atransmitter in order not to miss photons, which means that most of thedetector bandwidth is unused.

Most systems associated with a downstream implementation can beovercome. The most valuable resource should be shared by all users andshould operate at full capacity. One can build an upstream quantumaccess network in which the transmitters are placed at the end userlocation and a common receiver is placed at the network node. This way,an operation with up to 64 users is feasible, which can be done withmulti-user quantum key distribution over a 1×64 passive opticalsplitter.

The above described QKD scheme is applicable to twisted pair, coaxialcable, fiber optic, RF satellite, RF broadcast, RF point-to point, RFpoint-to-multipoint, RF point-to-point (backhaul), RF point-to-point(fronthaul to provide higher throughput CPRI interface forcloudification and virtualization of RAN and cloudified HetNet),free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as apersonal device cable replacement, RF and FSO hybrid system, Radar,electromagnetic tags and all types of wireless access. The method andsystem are compatible with many current and future multiple accesssystems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE. The techniques would be usefulfor combating denial of service attacks by routing communications viaalternate links in case of disruption, as a technique to combat TrojanHorse attacks which does not require physical access to the endpointsand as a technique to combat faked-state attacks, phase remappingattacks and time-shift attacks.

Thus, using various configurations of the above described orbitalangular momentum processing, multi-layer overlay modulation, and quantumkey distribution within various types of communication networks and moreparticularly optical fiber networks and free-space optic communicationnetwork, a variety of benefits and improvements in system bandwidth andcapacity maybe achieved.

Referring now to the drawings, and more particularly to FIG. 59, thereis illustrated one embodiment of a beam for use with the system. A lightbeam 5904 consists of a stream of photons 5902 within the light beam5904. Each photon has an energy ±

and a linear momentum of ±

k which is directed along the light beam axis 5906 perpendicular to thewavefront. Independent of the frequency, each photon 5902 within thelight beam 5904 has a spin angular momentum 5908 of ±

aligned parallel or antiparallel to the direction of light beampropagation. Alignment of all of the photons 902 spins gives rise to acircularly polarized light beam. In addition to the circularpolarization, the light beams also may be processed to carry an orbitalangular momentum 5910 which does not depend on the circular polarizationand thus is not related to photon spin.

Referring now to FIGS. 60 and 61, there are illustrated plane wavefrontsand helical wavefronts. Ordinarily, laser beams with plane wavefronts6002 are characterized in terms of Hermite-Gaussian modes. These modeshave a rectangular symmetry and are described by two mode indices m 6004and n 6006. There are m nodes in the x direction and n nodes in the ydirection. Together, the combined modes in the x and y direction arelabeled HG_(mn) 6008. In contrast, as shown in FIG. 61 beams withhelical wavefronts 6102 are best characterized in terms ofLaguerre-Gaussian modes which are described by indices I 6103, thenumber of intertwined helices 6104, and p, the number of radial nodes6106. The Laguerre-Gaussian modes are labeled LG_(mn) 610. For l≠0, thephase singularity on a light beam 6004 results in 0 on axis intensity.When a light beam 6004 with a helical wavefront is also circularlypolarized, the angular momentum has orbital and spin components, and thetotal angular momentum of the light beam is (l±

) per photon.

Existing techniques for improving electron hole recombination havevaried depending upon the particular type of device being utilized. Forexample as illustrated in FIG. 62, within biological light harvestingcomplexes (LHCs), a light beam 6202 is applied to the LHC such thatphotonic absorption 6204 creates molecular excited states. Thesemolecular excited states comprise excitons 6206 that are funneled toreaction centers 6208 where they are disassociated into electrons 6210and holes 6212. The electrons 6210 and holes 6212 are separated furthervia a series of cascading charge transfer steps.

As illustrated in FIG. 63, organic photovoltaic cells (OPVs) receive alight beam 6302 and use photo absorption 6304 to create the excitons6306 that are applied to a single donor-acceptor heterojunction 6308formed within a de-mixed blend of a donor and acceptor semiconductor togenerate the electrons 6310 and holes 6312. The most efficient OPVsystems comprise nanoscale (less than symbol 5 nm) domains of purefullerene acceptor and domains of fullerene intimately mixed with apolymer donor. These length scales are smaller than the Coulomb captureradius (CCR) in organic semiconductors (kT=e²4πε₀εr) which is estimatedto be as large as 16 nm at room temperature due to the low dielectricconstant of the material (approximately 3-4). Thus, in contrast to LHCs,electrons and holes diffusing through an OPV may encounter one anotherbefore they reach the electrodes. This is similar to standard inorganicsolar cells, where bimolecular electron-hole recombination (BR)determines solar cell performance. In addition to applying thesetechniques to OPV systems, the described techniques can be used forcontrolling electrons in solid state semiconductor materials, quantumcomputing systems, and biological systems as described below.

The rate of electron hole encounters that produce Coulombically boundstates is given by R=γn(p), where n(p) is the electron (hole) populationdensity and γ is the Langevin recombination constant given by γ=μ>/ε,where q equals the electric charge, <μ> equals the effectiveelectron/hole mobility and e equals the dielectric constant. This modelsuccessfully describes the principal operating mechanism of OLEDs, wherecharges injected through the electrodes capture one another to formstrongly bound excitons.

In empirically optimized OPV's, the recombination rate is suppressed byup to three orders of magnitude compared to the Langevin rate, allowingexternal quantum efficiencies as high as 80 percent. The recombinationof bound states formed via electron/hole encounters is mediated not onlyby energetics, but also by spin and delocalization, allowing for freecharges to be reformed from these bound states thus suppressingrecombination. This is more particularly illustrated in FIG. 64 whichillustrates when a light beam 6402 has spin and delocalizationcharacteristics 6404 applied thereto, the light beam including spincharacteristics 6406 may be applied to a photonic device 6408 (or otherdevices/systems). Results have shown that the photonic device 6408 (orother devices/systems) will have an improved suppression of therecombination rate due to the spin and delocalization characteristics.

Referring now also to FIG. 65, a similar suppression of recombinationsbetween electrons and holes may be achieved using twisted photons thatcarry orbital angular momentum. The light beam 6502 has an orbitalangular momentum applied thereto by for example a passive hologram, SLM(spatial light modulator), phase plate, etc. The generated twisted beam6506 may then result in new quantum states which slow down the rate ofrecombination when applied to a photonic device 6508 (or otherdevices/systems) providing further improved recombination rates.

A new design for an artificial photo-conversion system uses circuitry,such as a phase mask holograms, to apply orbital angular momentum (OAM)to the light signal and enable the suppression of electron holerecombination by avoiding the formation of triplet states in enhancingfluorescence efficiency. By placing OAM circuitry within the path of aphoton, the orbital angular momentum generated by the photon can betransferred to an electron and a new quantum state created wheresuppression of electron-hole recombination is supported. Thissuppression is due to the change in total angular momentum of theelectron (spin+orbital) using a device that twists the protons with aprescribed topological charge using a variety of methods (passivelyusing a hologram or actively using other methods). The amount of orbitalangular momentum applied results in a specific topological charge thatcan control the rate of recombination. Thus, by controlling the amountof applied OAM, the rate of suppression of recombination may becontrolled.

Thus, referring now to FIG. 66, there is illustrated a functional blockdiagram of one manner for improving the suppression of electron holerecombination in a photonic circuit. A light beam 6602 is passed throughan OAM device 6604 in order to generate an OAM twisted light beam 6606.The OAM device 6604 may comprise a passive hologram, a spatial lightmodule (SLM), a spatial plate, an amplitude mask, phase mask or anyother device capable of applying an orbital angular momentum to thelight beam. The amount of OAM twist applied to a light beam may becontrolled via an OAM control circuit 6605. The OAM control circuit 6605may control the level of OAM twist based upon a desired level ofelectron-hole recombination within a photonic device. The OAM twistedlight beam 6606 is applied to some type of photonic circuit 6608 thatenergizes electrons responsive to the OAM light beam 6606 to higherstates that will decompose into a ground state. The photonic circuit maycomprise optical LEDs, organic photovoltaic cells, biological lightharvesting complexes, organic semiconductors, PN junctions,semiconductor, solid state devices, solar cells or any other lightaffected device. Additional components such as semiconductors, quantumcircuits and biological materials may also have OAM applied to theirelectrons from an OAM light beam.

Referring now to FIG. 67, there is illustrated the manner in whichelectrons move from different states between various quantum stateswithin an organic photovoltaic cell or other semiconductor, quantumcircuit, or biological material. Conversions between excited statespecies are shown at 6702 while recombination channels are shown at6704. State S₁ 6706 and state T₁ 6708 are the lowest lying singletexcitons 6706 and triplet excitons 6708, respectively. CT is the chargetransfer state. Photoexcitation goes from the ground state S₀ 6710 to asinglet excitation S₁ 6706 at 6712. The singlet exciton S₁ 6706 isionized at a heterojunction leading to the formation of ¹CT states whichseparates into free charges (FC) 6702, 6704 with high efficiency at6714. Biomolecular recombination of electrons and holes leads to theformation of ¹CT state at 6716 and ³CT states at 6718 in a 1:3 ratio asmandated by spin statistics. The ¹CT can recombine slowly to the groundstate as shown at 6720. The ³CT state recombination to the ground stateS₀ 6710 is spin forbidden but relaxation of the T₁ state 6708 at 6722 isenergetically favorable. Once formed triplet excitons ³CT can return toground state via an efficient triplet charge annihilation channel at6724. Under favorable conditions, the time required for CT states toreorganize to free charges at 6718 is less than the time required forrelaxation to T₁ at 6722. Thus, CT states are recycled back to freecharges leading to a suppression of recombination.

In order to probe the dynamics of these bound states, we can firstconsider the initial dissociation of the photo generated singletexciton, S₁ 6706 at the D-A interface. The first step of this process ischarge transfer across the D-A interface, which can lead to eitherlong-range charge separation or the formation of bound interfacialcharge transfer (CT) states. Such bound charge pairs then decay to theground state S₀ 6710 via geminate recombination (GR). It is important tonote that spin must be taken into account when considering CT states asthey can have either singlet (¹CT) or triplet (³CT) spin character whichare almost degenerate in energy. Disassociation of photo generatedsinglet excitons leads to the formation of only ¹CT states 6704 due tospin conservation. In contrast, recombination of spin-uncorrelatedcharges leads to the formation of ¹CT and ³CT states in a 1:3 ratiobased on spin statistics. ¹CT states can either dissociate or recombineto the ground state either via luminescence which is slow for thisintermolecular D-A process or non-radioactive decay. For ³CT states,decay to ground states is spin forbidden and hence both radiative andnon-radiative processes are very slow. However, if the energy of thelowest lying molecular triplet exciton (T₁) lies below the ³CT energy,then ³CT can relax to T₁.

The model for recombination in the importance of spin statistics arewell-established in OLEDs where the formation of non-luminescent tripletexcitons is a major loss mechanism. Efforts to overcome this problemhave focused on the use of metal organic complexes to induce spinorbiting coupling and recently on the use of low exchange energymaterials that can promote inner system crossing from T₁ to S₁.

Thin films using transient absorption (TA) spectroscopy have been usedin the past. In this technique a pump pulse generates photoexcitationswithin the film. At some later time, the system is interrogated using abroadband probe pulse. Although TA has been widely employed to study thephotophysics of OPV blends previous measurements have been severallylimited by three factors. The first limitation has an insufficienttemporal range, typically a maximum of 2 ns delay between pump andprobe. A second limitation has a limited spectral range and lack ofbroadband probes, which hinders the observation of dynamic interactionsbetween excitations. And lastly, insufficient sensitivity, whichmandates the use of high fluence pump pulses to create large signals.

These issues have been resolved recently using broad temporal (up toIms) and spectral windows (out to 1500 nm) and high sensitivity (betterthan 5×10-6). This temporal window is created by using an electricallydelayed pump-pulse and allows for the study of long-lived charges andtriplet excitons. In conjugated polymers local geometrical relaxationaround charges (polaron formation) causes rearrangement of energylevels, bringing states into the semiconductor gap and giving rise tostrong optical transitions 700 nm-1500 nm. The absorption bands ofsinglet and triplet excitons are also found to lie in the near IR makinga broadband spectral window necessary to track the evolution of theexcited state species. The high sensitivity of the experiment isessential as it allows one to probe the dynamics of systems when theexcitation densities are similar to solar illumination conditions(10¹⁶-10¹⁷ excitations/cm). At higher excitation densities bimolecularexciton-exciton and exciton-charge annihilation processes can dominate,creating artifacts, making such measurements unreliable indicators ofdevice operation. One can further combine these measurements withadvanced numerical techniques that allow one to resolve the spectralsignatures of the overlapping excited state features and track theirkinetics.

The overlapping spectrum of the excited states makes the analysis oftheir kinetics difficult. In order to overcome this problem one can usea genetic algorithm (GA), which enables us to extract the individualspectra and kinetics from the data set. Within this approach a linearcombination of two or more spectra and associated kinetics can be takenand ‘evolved’ until they best fit the experimental data.

The extracted kinetics can demonstrate that triplets may grow as chargesdecay. One can consider that the primary decay channel for triplets istriplet-charge annihilation, due to the high charge densities present,and model the time evolution of the system with the Langevin equationgiven below:

$\frac{dN_{T}}{dt} = {{{- a}\frac{dp}{dt}} - {{\beta\left\lbrack N_{T} \right\rbrack}\lbrack p\rbrack}}$where:p: Charge concentration;N_(T): Triplet concentration;a: is the fraction of decaying charges that form triplets;β: is the rate constant for triplet-charge annihilation.

We now turn to the question of whether the time taken for relaxationfrom ³CT to T₁, process 6722 shown in FIG. 67 with an associatedtimescale τ₄, is fast and if not, whether there are the competingprocesses for the decay of ³CT. As noted earlier the CT energy liesabove T₁, making relaxation from ³CT to T₁ energetically favored.However, for the more efficient 1:3 blend, no triplet formation ispossible at room temperature. But at low temperatures (<240K),bimolecular triplet formation can be observed in this blend. Thissuggests that there is a thermally activated process that competes withrelaxation to T₁. We consider this process to be the dissociation of ³CTback to free charges. Thus at high temperatures (>240K) the dissociationof ³CT back to free charges, process 6718 shown in FIG. 67 with anassociated timescale τ₃, out competes relaxation of ³CT to T₁ i.e.τ₄>τ₃. Hence one of the two channels for recombination (the other beingrecombination through ¹CT) is suppressed, allowing for high EQEs. Atlower temperatures this dissociation process is suppressed, such thatτ₄<τ₃, leading to a buildup of triplet excitons.

Referring now to FIG. 68, there is illustrated a flow diagram describingone manner for utilizing the application of orbital angular momentum toa light beam to suppress electron-hole conversion within a photonicdevice. Initially at step 6810, the desired OAM level is selected forapplication to the light beam. The selected OAM level will provide adesired suppression of the electron-hole recombination rate. The OAMgenerating device is placed in the path between the light beam and thephotonic device at step 6812. The light beam is allowed to pass at step6814 through the OAM generating device and the OAM twist is applied tothe light beam. The OAM twisted light beam then interacts with thephotonic device to generate energy at step 6816 such that theelectron-hole recombination within the photonic device is affected bythe OAM twisted beam.

OAM Beam Interactions with Other Matter

As mentioned above, OAM infused light beams may also interact with othertypes of matter such as semiconductors, quantum circuits and biologicalmaterials. Light can carry both spin and orbital angular momentums.Multiple patents describing applications of OAM in several areasincluding communications, spectroscopy, radar and quantum informaticssuch as U.S. patent application Ser. No. 14/882,085, entitledAPPLICATION OF ORBITAL ANGULAR MOMENTUM TO FIBER, FSO AND RF, filed onOct. 13, 2015; U.S. patent application Ser. No. 16/226,799, entitledSYSTEM AND METHOD FOR MULTI-PARAMETER SPECTROSCOPY, filed on Dec. 20,2018; U.S. patent application Ser. No. 16/509,301, entitled UNIVERSALQUANTUM COMPUTER, COMMUNICATION, QKD SECURITY AND QUANTUM NETWORKS USINGOAM QU-DITS WITH DLP, filed on Jul. 11, 2019, each of which isincorporated herein by reference in its entirety. However, thesetechniques may be used to extend the applications of OAM to solid statesystems including potential techniques to improve the efficiency ofsolar cells, display units and other applications.

As described above, quantum technology uses individual atoms, molecules,and photons to construct new kinds of devices. Quantum objects are notlimited to the binary rules of conventional computing but can be in twoor more logical states at once. Therefore, it is potentially possible toconstruct a higher-radix quantum system. When a group of objectscollectively occupy two or more states at once, they are said to beentangled. Photons interact very weakly with their environment, andtherefore entangled states of photons hold considerable promise forapplications ranging from imaging and precision measurement tocommunications and computation.

Quantum entanglement is a physical phenomenon that occurs when pairs orgroups of particles are generated, interact, or share spatial proximityin ways such that the quantum state of each particle cannot be describedindependently of the state of the others, even when the particles areseparated by a large distance.

An entangled system is defined to be one whose quantum state cannot befactored as a product of states of its local constituents; that is tosay, they are not individual particles but are an inseparable whole. Inentanglement, one constituent cannot be fully described withoutconsidering the other(s). The state of a composite system is alwaysexpressible as a sum, or superposition, of products of states of localconstituents; it is entangled if this sum necessarily has more than oneterm. Quantum systems can become entangled through various types ofinteractions. For some ways in which entanglement may be achieved forexperimental purposes. Entanglement is broken when the entangledparticles decohere through interaction with the environment; forexample, when a measurement is made.

As an example of entanglement: a subatomic particle decays into anentangled pair of other particles. The decay events obey the variousconservation laws, and as a result, the measurement outcomes of onedaughter particle must be highly correlated with the measurementoutcomes of the other daughter particle (so that the total momenta,angular momenta, energy, and so forth remains roughly the same beforeand after this process). For instance, a spin-zero particle could decayinto a pair of spin-½ particles. Since the total spin before and afterthis decay must be zero (conservation of angular momentum), whenever thefirst particle is measured to be spin up on some axis, the otherparticle, when measured on the same axis, is always found to be spindown. (This is called the spin anti-correlated case; and if the priorprobabilities for measuring each spin are equal, the pair is said to bein the singlet state.)

In every quantum-information processing scheme, bits of information mustbe stored in the states of a set of physical objects, and there must bea physical means to affect an interaction between these quantum bits(qubits or qudits for higher-radix systems as described herein above).But when individual photons are used to store and process information, aproblem shows up as photons interact extremely weakly or not at all, andit might thus seem unlikely that they can be used directly forquantum-information processing.

A scheme for effective quantum interactions between photons has beenconstructed that does not require the photons to interact physically.When a group of photons are in an entangled state, the measurement ofone or more of these photons' states causes the state of the remainingphotons to collapse. Such a state collapse can be viewed as an effectiveinteraction between the remaining photons. This collapse can be used toget photons to interact and entangle. The nature of the interactiondepends on the outcomes of the measurement on each observed photon. Theoccurrence of certain predetermined outcomes of the measurements signalsthe effective interaction of the remaining photons.

This approach can be used to implement quantum logic gates in a schemecalled linear-optical quantum computing. Substantial progress towardlinear-optical quantum computing has been made, such as the design anddemonstration of a teleportation gate and of a controlled-NOT(C-NOT)logic gate, and the preparation of entangled states of twophotons. But serious challenges remain.

When using photons as qubits to encode information, each photon must becreated in one of two distinct quantum states or in a superposition ofthese two states. If the photons occupy more states in an uncontrolledway, the scheme for effecting interactions among photons will fail. Thequantum states of a photon are characterized by their spectrum, pulseshape, time of arrival, transverse wave vector, and position. All thesevariables must be controlled precisely to ensure that only two distinctquantum states are present and to perform.

Since photons do not interact with one another well, a technique isdescribe that provides a way to have photons interact with various typesof quantum matter and control the quantum transitions in thesemiconductor material to build quantum gates for quantum computing. Toexpand the capabilities beyond 2-states qubits, OAM infused beam oftwisted light may be used in order to create higher-radix systems thatcould represent a qudit (not qubits) using the techniques describedabove.

First, a theoretical framework for transitions between quantized statesof a material (i.e. optical transitions with light frequencies above thebandgap), so that free carriers rather than excitons are produced isconstructed as controlled by the level of OAM within an applied lightbeam using an associated control circuit. There is a transfer of angularmomentum between the light and the photo-excited electrons so that a netelectric current is created. A magnetic field can also be induced bythese photocurrents. This is more particularly illustrated in FIGS. 69and 70. FIG. 69 illustrates this process wherein light beam generationis carried out at 6902 to provide a light beam 6903 to an OAM generationprocess 6904. The OAM generation process 6904 applies an orbital angularmomentum to the plane wave light in the manner described hereinabove inorder to provide an OAM processed beam 6905. The OAM processed beam 6905is applied to physical matter to cause the OAM excitation of electronswithin the matter at 6906. The OAM processed beam 6905 having OAM withinthe photons of the transmitted light team transfers the OAM to theelectrons of matter at the process 6906 generating the electric currentand magnetic field.

The structure for carrying out this process is more particularlyillustrated in FIG. 70. A light beam generator 1002 generates the lightbeam 6903. The light beam 6903 comprises a plane wave light beam asdescribed hereinabove, and is applied to the OAM generator 7004 suchthat an orbital angular momentum may be applied to the plane wave lightbeam imparting OAM values to the photons of the light beam 6903. The OAMgenerated beam 6905 is provided to some type of beam transmitter 7006that transmits the light beam 6905 toward or at a particular physicaldevice/system 7008. The physical device/system 7008 may comprise varioustypes of semiconductor materials, quantum computer components,biological materials and solid-state materials.

Referring now to FIG. 71, semiconductors 7102 are solids which at zerotemperature have the highest occupied (valence) energy bands 7104 andlowest empty (conduction) energy bands 7106 separated by a gap E_(g)7108. In this respect, they are closer to insulators than to metals.However, in typical semiconductors E_(g) ≃1 eV, making possible thetransitions between the valence band 7104 and conduction band 7106 byoptical excitation 7110. In a crystalline semiconductor, electrons inthe valence band 7104 and conduction band 7106 occupy the Bloch states.This simplified 2-band model can have the main features of asemiconductor and is a good model of a real bulk system. A frameworkdescribes the inter-band transitions in a 2-band model of a bulksemiconductor induced by twisted light and is further described hereinbelow. The coherent optical excitation to conduction-band states (thecase with photon energy> bandgap energy) must be evaluated. In thiscase, the creation of excitons is negligible, and the free carrierstransferred by optical excitation from the valence band 7104 to theconduction band 7106 can be considered. The process is performed byfirst constructing the complete semi-classical Hamiltonian operatorswhere quantum mechanical operators are used for electrons and classicalvariables are used for the light field consisting of two terms, the bareelectron energy and the interaction Hamiltonian.

In order to understand the effect that a twisted-light beam has on theground state of a semiconductor, we look at the positive part of theinteraction Hamiltonian. The action of this interaction Hamiltonian onthe full ground state of the N-electron semiconductor provides aneigenvector with expectation value of the orbital angular momentum equalto h.

Referring now to FIG. 72, the net transfer of orbital angular momentumfrom photons 7202 to the photo-excited electrons 7204 causes an electriccurrent 7206 and associated magnetic field 7208. They both may bedetected, but certainly the magnetic field 7208. These detectablephenomena of optical excitation may also result in opto-electronicapplications.

Referring now to FIG. 73, there is illustrated a flow diagram of theprocess for determining a total current/magnetic field provided by agroup of electrons. An estimate of the total current/magnetic fieldproduced by all electrons can be obtained by calculating the totalnumber of electrons excited by the field at step 7302, and thendetermining at step 7304 the current/magnetic field of a singleelectron. Finally, multiplying at step 7306 the single-electroncurrent/magnetic field by the number of photo-excited electrons. Startwith a single particle framework which does not consider theelectron-electron Coulomb interaction. This assumption is acceptablebecause the main correction introduced by the Coulomb interaction wouldbe the renormalization of the single-particle energies. This mean-fieldeffect would be small due to the case of low excitation (the density ofphoto-generated electrons and holes is small), and it does not affectthe model of the framework. Beyond the mean-field approximation, theCoulomb interaction introduces complex and possibly interestingscattering effects.

The excitation process generates a superposition of conduction-bandstates with a well-defined angular momentum. This superposition state isassociated to an electric-current density, which acts as the source of amagnetic field. Note that a single-particle theory has the drawback ofnot considering the Pauli exclusion in the photoexcitation of multipleelectrons and it leaves out the electron-electron interaction effects.Thus, a more complete theoretical treatment of the inter-band excitationby twisted light of solids is needed.

The complete framework should include a twisted-light OAM-generalizedsemiconductor Bloch equations from the Heisenberg equations of motion ofthe populations and coherences of the photo-excited electrons. Thisframework would be valid for pulsed or CW twisted-light beams andconsiders the inhomogeneous profile of the beam, as well as the transferof momentum from the light to the electrons in the plane perpendicularto the beam's propagation direction. If excitonic phenomena are nottargeted, the Coulomb interaction does not play a major role in thebasic physics of band-to-band optical transitions (except the case ofexcitons), and for that reason the framework to a free-carrierformulation of the theory is limited. Referring now to FIG. 74, as afirst step one can derive and solve, first the mean-field for twistedlight at step 7402, and later the same equations with collision terms atstep 7404. The mean-field for twisted light is critical which can bemodified with collusion terms. Collision terms describe the scatteringprocesses that happen by the collisions of photo-excited electrons,(electron-electron, electron-phonon, and electron-impurity scattering).Collision terms in the relaxation-time-approximation can be added to theframework in order to describe those scattering processes, and anumerical solution of the resulting equations of motion would allow usto explore the influence of collisions on the effects at step 7406.Also, note that while the discussion focuses on bulk systems, theframework can easily be applied to two-dimensional systems excited atnormal incidence.

Usually, the optical excitation in bulk systems is theoretically dealtwith by assuming that the system is a cube, quantizing the electronsusing Cartesian coordinates and taking, at the right moment in thederivation, the limit of large system size. For symmetry reasons, thismethod allows straightforward calculations in the case of excitationwith plane-wave light. However, it leads to a complex formulation in thecase of excitation by twisted light having OAM applied thereto. Atwisted light beam has an inherently cylindrical nature and that is thecoordinate system used for our formulation.

In the framework for bulk systems, this simple but key idea is takenadvantage of for calculations as shown in FIG. 75. The solid that alight beam interacts with is imagined as a cylinder at step 7502. Theelectron states are quantized in cylindrical coordinates at step 7504.Finally, the limit of a large system is taken at step 7506. Bulkproperties are independent of the geometry of the solid. The requiredmicroscopic structure of the Bloch wave functions is kept in order tocharacterize the valence and conduction band states: the periodic partsof the Bloch states are approximated by their values at zero crystalmomentum, a common practice known as effective-mass approximation. Theuse of cylindrical rather than Cartesian coordinates allows decouplingof the Heisenberg equations of motion according to values of theelectron angular momentum, which greatly reduces the complexity of theproblem. Using these generalized Bloch states one can predict thekinetics of electrons, look for the occurrence of electric currents withcomplex profiles, and demonstrate the transfer of OAM from the lightbeam to the electrons.

In this framework, a consideration of a direct-gap semiconductor orinsulator and attempt to formulate inter-band transitions caused byillumination with a beam of twisted light may be made. Assume that thelight's frequency is such that mainly band-to-band transitions occur sothat exciton creation is unimportant. Under this regime it issatisfactory to formulate a theory not including the Coulomb interactionbetween carriers. Thus, the framework can potentially describe theelectrons' kinetics, from irradiation to a fraction of picoseconds toavoid strong deviation due to decoherence. The framework is applicableto many physical systems, from semiconductors having bandgap of afraction of an eV through several eV, up to insulators having largergaps, provided that the frequency of the twisted field is tuned abovethe energy bandgap. This requires the use of twisted fields in the nearinfrared to UV spectrum. The process describes a system and method forformulating the transfer of orbital angular momentum between the twistedlight and the electrons and describing what the electron distributionlooks like as a result of the photoexcitation. Although several valencebands may be involved in inter-band optical transitions, here forsimplicity a 2-band model is considered. The generalization of thetheory to the case with more than one valence band involved isstraightforward from this 2-level framework.

The solutions to equations presented are the building blocks forconstructing mean values of observables of interest. In the standardtheory of optical transitions, where the light is assumed to be a planewave and the dipole approximation is made, once the time and momentumdependent density matrix is obtained, one calculates the macroscopicoptical polarization and from it, for example, the opticalsusceptibility. Under those assumptions, the macroscopic polarization isjust a spatially uniform, time-dependent, function.

By contrast, if the inhomogeneities of the field are considered (e.g.finite beam waist and oscillatory dependence in the propagationdirection), the electronic variables acquire an interesting spacedependence. The excitation of solids by twisted light beams alsoproduces a space-dependent carrier kinetics which requires localvariables for its description. It is possible to visualize the patternof motion of the photo-excited electrons by calculating and using thespatially inhomogeneous electric current density. Another usefulvariable, the transferred angular momentum, is instead a globalmagnitude that characterizes the twisted light matter interaction. Onecan study separately the contributions to the angular momentum and theelectric current made by the inter-band coherences, on the one hand, andby the populations and intra-band coherences, on the other hand. Thisseparation is conceptually useful because the inter-band coherencecontributions have fast (perhaps femtosecond) oscillations around a nullvalue of the current or angular momentum, analogously to what happenswith the inter-band polarization, while the population or intra-bandcoherence contributions come from slower (perhaps picosecond) processesin which a net transfer of momentum from light to matter occurs.However, inter-band coherence contributions are slower processes wherewe have a transfer of momentum from light to matter. The latter arerelated to the photon-drag effect which is now generalized toincorporate a rotational drag in the plane perpendicular to thepropagation direction, due to the “slow” transfer of angular momentum.

The mathematical model begins with Schrödinger equation which comprises:

|Ψ

=E|Ψ

wherein

: Hamiltonian for total Energy operator defined by:

→T+Vand T comprises the kinetic energy and V is the potential energy.The classical representation of the Hamiltonian for total energy is:

$H = {\frac{p^{2}}{2m} + V}$where

$\frac{p^{2}}{2m}$is the kinetic energy and V is the potential energy.

For the simplest interaction (Hydrogen):

${V(r)} = {{- \frac{{ze}^{2}}{4\;\pi\;\epsilon_{0}r}}{Coulomb}}$${p->{{{- i}\;\hslash{\nabla\Psi}}->{\Psi\left( \overset{->}{r} \right)}}} = {\Psi\left( {r,\theta,\phi} \right)}$$H = {\frac{- \hslash^{2}}{2m}{\nabla^{2}{- \frac{{ze}^{2}}{4\;{\pi\epsilon}_{0}r}}}}$Using reduced masses

$\left. m\rightarrow\mu \right. = {\frac{m_{e}m_{p}}{m_{e} + m_{p}}.}$A 2-body problem can be solved as one body problem with reduced mass.Thus:

${{\nabla^{2}{\Psi\left( \overset{\rightarrow}{r} \right)}} + {\frac{2\mu}{\hslash^{2}}\left( {E + \frac{{ze}^{2}}{4{\pi\epsilon}_{0}r}} \right){\Psi\left( \overset{\rightarrow}{r} \right)}}} = 0$

The Laplacian can be provided in spherical coordinates as follows:

${\left. \nabla^{2}\rightarrow \right.\frac{1}{r^{2}}{\frac{d}{dr}\left( {r^{2}\frac{d}{dr}} \right)}} + {\frac{1}{r^{2}\sin\;\theta}\frac{d}{d\;\theta}\left( {\sin\;\theta\frac{d}{d\theta}} \right)} + {\frac{1}{r^{2}\sin^{2}\theta}\frac{d^{2}}{d\;\phi^{2}}}$Assuming:Ψ_(nlm)(r,θ,ϕ)=R _(nl)(r)Y _(lm)(θ,ϕ)wherein R_(nl)(r) comprises the radial portion and Y_(lm)(θ,ϕ) comprisesthe angular portionThe radial portion is solved according to:

${R_{nl}(r)} = {\sqrt{\left( \frac{2}{na_{0}} \right)^{3}\frac{\left( {n - l - 1} \right)!}{2{n\left\lbrack {\left( {n + l} \right)!} \right\rbrack}^{3}}}{e^{- \frac{r}{{na}_{0}}}\left( \frac{2r}{na_{0}} \right)}^{l}{\mathcal{L}_{n - l - 1}^{{{2l} +},i}\left( \frac{2r}{na_{0}} \right)}}$The angular portion is solved according to:

${Y_{l\; m}\left( {\theta,\phi} \right)} = {\left( {- 1} \right)^{m}\sqrt{\frac{\left( {{2l} + 1} \right){\left( {l - m} \right)!}}{4\;{{\pi\left( {l + m} \right)}!}}}{P_{l\; m}\left( {\cos\;\theta} \right)}e^{{im}\;\phi}}$$a_{0} = \frac{\hslash^{2}}{\mu\; e^{2}}$

Now E₂−E₁=

ω transitions between two quantized states and the transition rate isdefined according to:

${{transition}\mspace{14mu}{rate}} = \frac{{probability}\mspace{14mu}{of}\mspace{14mu}{transition}}{{unit}\mspace{14mu}{time}}$The transition rate between two states is proportional to the strengthof coupling between two states including initial state (i) and finalstate (f). Fermi's golden rule is defined as:

$W_{if} = {\frac{2\pi}{\hslash}{M_{if}}^{2}{\rho(E)}}$where M_(if)=

i|H|f

=∫Ψ_(f)*(r)H(r)Ψ_(i) (r)d³r

Using a semi-classical approach for interaction of light-matter whereinatoms are treated quantum mechanically and photons are treatedclassically, the QED Hamiltonian is

=H _(radiation) +H _(matter) +H _(interaction)Wherein:

$H_{rad} = {\frac{\epsilon_{0}}{2}{\int{\left( {E^{2} + {c^{2}B^{2}}} \right)d^{3}r}}}$H_(matter)=Schrodinger HamiltonianH _(int) =−P·∈ ₀H_(int) comprises a dipole transition wherein −P equals an atom electricdipole and ∈₀ comprises the light electric field.

For an atom electric dipole:

$P = {\sum\limits_{i}{q_{i}r_{i}}}$For a single electron:P=−er

In the context of a semiconductor, the Bohr radius a₀ is the distancebetween electron and hole of a bound exciton. The Hamiltonian forexciton state:

$H = {{- \frac{\hslash^{2}}{2M}}{\nabla_{e}^{2}{- \frac{\hslash^{2}}{2\mu}}}{\nabla_{h}^{2}{- \frac{e^{2}}{\epsilon\left( {r_{e} - r_{h}} \right)}}}}$M = m_(e)^(*) + m_(h)^(*)$\mu = \frac{m_{e}^{*}m_{h}^{*}}{m_{e}^{*} + m_{h}^{*}}$

With similar solutions R_(nl)(r) and Y_(lm)(θ, ϕ) with

$a_{ex} = {a_{0}\epsilon\frac{m_{0}}{\mu}}$(m_(o) is the free electron mass) instead of a₀.

In a box of length L, the quantized eigen energies are:

$E_{n} = {\frac{n^{2}\hslash^{2}\pi^{2}}{2\; m\; L^{2}} = \frac{n^{2}h^{2}}{8\; m\; L^{2}}}$

Using exciton Hamiltonian above:

${\Delta E} = {E_{gap} + {\frac{h^{2}}{8R^{2}}\left\lbrack {\frac{n_{e}^{2}}{m_{e}^{*}} + \frac{n_{h}^{2}}{m_{h}^{*}}} \right\rbrack}}$

Now from OAM of light it is known that:A(r,t)={circumflex over (π)}_(±E(r,ϕ,z)e) ^(i(kz-ωt))where {circumflex over (π)}_(±) comprises a polarization vector definedas:

${\overset{\hat{}}{\pi}}_{\pm} = \frac{\overset{\hat{}}{x} \pm {i\overset{\hat{}}{y}}}{\sqrt{2}}$A(r, t) = A_(a)(r, t) + A_(e)(r, t)where A_(a)(r, t) comprises the absorption from the valence band to theconduction band (v→c) and A_(e)(r, t) comprises the emissions from theconduction band to the valence band (c→v)Next with paraxial approximation:

${{\nabla^{2}{E(r)}} + {2ik\frac{d}{dz}{E(r)}}} = 0$

With LG-beam solution from a cylindrical Laplacian:

${E_{lP}\left( {r,\phi,z} \right)} = {{\frac{C_{lp}}{w(z)}\left\lbrack \frac{\sqrt{2}r}{w(z)} \right\rbrack}^{l}e^{- \frac{r^{2}}{w^{2}{(z)}}}{\mathcal{L}_{lP}\left( \frac{2\; r^{2}}{w^{2}(z)} \right)}e^{\lbrack{{- \frac{{ikr}^{2}z}{2{({z^{2} + z_{R}^{2}})}}} - {{il}\;\phi} + {{i{({{2\; P} + l + 1})}}{ta}\; n^{- 1}\frac{z}{z_{R}}}}\rbrack}}$$\mspace{20mu}{c_{lp} = {\left( {- 1} \right)^{P}\sqrt{\frac{2\;{P!}}{{\pi\left( {{l} + P} \right)}!}}}}$  z_(R) = Rayleigh  length  typically  several  meters$\mspace{20mu}{{\mathcal{L}_{lP}(x)} = {\frac{x^{- l}e^{x}}{n!}\frac{d^{P}}{{dx}^{P}}\left( {x^{P + l}e^{- x}} \right)}}$Where z_(R)>>z

$\lambda = \frac{2\;\pi}{\sqrt{k_{z}^{2} + k_{t}^{2}}}$$\omega = {c\sqrt{k_{z}^{2} + k_{t}^{2}}}$

For z_(R)>>z approximation

w(z) → w₀$\left. \frac{{kr}^{2}z}{2\left( {z^{2} + z_{R}^{2}} \right)}\rightarrow 0 \right.$$\left. {\left( {{2P} + l + 1} \right)\tan^{- 1}\frac{Z}{Z_{R}}}\rightarrow 0 \right.$

Then

${{E_{lP}\left( {r,\phi,z} \right)} \cong {{\frac{c_{lP}}{w_{0}}\left\lbrack \frac{\sqrt{2}r}{w_{0}} \right\rbrack}^{l}e^{- \frac{r^{2}}{w_{0}^{2}}}{\mathcal{L}_{lP}\left( \frac{2\; r^{2}}{w_{0}^{2}} \right)}e^{{- {il}}\;\phi}}} = {{f(r)}e^{{- {il}}\;\phi}}$where f(r) comprises

${\frac{c_{lP}}{w_{0}}\left\lbrack \frac{\sqrt{2}r}{w_{0}} \right\rbrack}^{l}e^{- \frac{r^{2}}{w_{0}^{2}}}{{\mathcal{L}_{lP}\left( \frac{2r^{2}}{w_{0}^{2}} \right)}.}$

Assume the wave function for electronic states of semi-conductorΨ(r,θ,ϕ)=Φ(r,ϕ)Z(z)With cell periodic structure of lattice defined by u_(b)(r) and spin ξΨ_(c)(r)=[Φ((r,ϕ)Z(z)]U _(c)(r)ξ; conduction-bandΨ_(v)(r)=[Φ(r,ϕ)Z(z)]U _(v)(r)ξ valence-bandΦ_(nm)(r,ϕ)=R _(nm)(r,ϕ)e ^(imϕ)Where

${R_{n\; m}\left( {r,\ \phi} \right)} = {\frac{\left( {- 1} \right)^{n}}{\sqrt{2\;\pi}L}\sqrt{\frac{n!}{\left( \left. {n +} \middle| m \right| \right)!}}{e^{\frac{r^{2}}{4{\mathfrak{L}}^{2}}}\left( \frac{r}{\sqrt{2}{\mathfrak{L}}} \right)}^{m}{\mathcal{L}_{n\; m}\left( \frac{r^{2}}{2{\mathfrak{L}}^{2}} \right)}}$and L is the characteristic confinement length of electrons.

We know that to put the interactionmechanical momentum→quntum momentum−qA; and

$H = \frac{P^{2}}{2m}$

So the single electron interaction Hamiltonian would be:

$H_{int} = {- {{\frac{1}{2\; m_{e}^{*}}\left\lbrack {\overset{->}{P} - {q\;{\overset{->}{A}\left( {r,t} \right)}}} \right\rbrack}\left\lbrack {\overset{->}{P} - {q\;{\overset{->}{A}\left( {r,t} \right)}}} \right\rbrack}}$q = −eP=−i

∇ operator

For minimal interaction:

$H_{int} = {{- \frac{q}{m_{e}^{*}}}{{\overset{->}{A}\left( {r,t} \right)} \cdot \overset{->}{P}}}$

Therefore the transition matrix from initial to final state is:conduction→valenceM _(if) =

i|H|f

=

cα′|H|vα

where α=collective index that includes all quantum numbers

$M_{if} = {\left\langle {i{H}f} \right\rangle = {{- \frac{q}{m_{e}^{*}}}\left\langle {\Psi_{{ca}^{\prime}}{{{A\left( {r,t} \right)} \cdot P}}\Psi_{v\;\alpha}} \right\rangle}}$$M_{if} = {\left\langle {i{H}f} \right\rangle = {{\frac{i\;\hslash\; q}{m_{e}^{*}}{\int{{\Psi_{c\;\alpha^{\prime}}^{*}\left( \overset{->}{r} \right)}{{\overset{->}{A}\left( {\overset{->}{r},t} \right)} \cdot {\overset{->}{\nabla}{\Psi_{v\;\alpha}\left( \overset{->}{r} \right)}}}d^{3}r}}} = {c_{1} + c_{2} + c_{3}}}}$Using ∇·(AB)=∇A·B+A(∇·B)

We haveΨ_(c)({right arrow over (r)})=Ψ_(c)(r,ϕ,z)=R _(nm)(r,ϕ)e ^(imϕ) Z(z)U_(c)(r)ξΨ_(v)({right arrow over (r)})=Ψ_(v)(r,ϕ,z)=R _(nm)(r,ϕ)e ^(imϕ) Z(z)U_(v)(r)ξwhere R_(nm)(r, ϕ)e^(imϕ) comprises Φ*(r,ϕ); U_(c)(r) comprisescell-periodic; and ξ comprise spin.

Due to orthogonality

_(c)|U_(v)

=δ_(cv)=∫U_(c)*({right arrow over (r)}) U_(v)({right arrow over (r)})d³r

  Ψ_(c)^(*) = Φ^(*)(r, ϕ)Z^(*)(z)U_(c)^(*)ξ_(c)^(*)  Ψ_(v) = Φ(r, ϕ)Z(z)U_(v)ξ_(v)$M_{if} = {{c_{1} + c_{2} + c_{3}} = {\frac{i\;\hslash\; q}{m_{e}^{*}}{\int\limits_{L^{3}}{{\Phi_{C}^{*}\left( \overset{\rightarrow}{r} \right)}{\Phi_{v}\left( \overset{\rightarrow}{r} \right)}{{\overset{->}{A}(r)} \cdot \left\lbrack {{U_{c}^{*}\left( \overset{\rightarrow}{r} \right)}{\nabla{U_{\nu}(r)}}} \right\rbrack}{{Z(z)}}^{2}\xi_{C}^{*}\xi}}}}$

With spin orthogonality ∫ξ′ξd³r=δ_(ξ′ξ)

Then the transition matrix for the conductance to valence transitioncomprises:M _(cv) =

cα′|H _(a) |vα

where H_(a) is the absorbtion.The matrix for the valence to conductance transition comprises:M _(vc) =

vα|H _(e) |cα′

where H_(e) is the matrix element.

${a^{3}P_{ij}} = {\int\limits_{a^{3}}{d^{3}r{U_{i}^{*}(r)}\left( {{- i}\hslash\nabla} \right){U_{j}(r)}}}$where P_(ij) is the matrix element.

The transmission matrices may be further defined as:

$M_{cv} = {{- {{ie}^{i\;\omega\; t}\left( \frac{2\pi\;\hslash\; q}{m} \right)}}\left( {\hat{\pi} \cdot P_{cv}} \right)\delta_{l,{({m^{\prime} - m})}}\delta_{\xi_{v}\xi_{c}}{\int{r\;{{drf}(r)}{R_{c\;\alpha^{\prime}}^{*}(r)}{R_{v\;\alpha}(r)}}}}$$M_{vc} = {{- {{ie}^{i\;\omega\; t}\left( \frac{2\pi\;\hslash\; q}{m} \right)}}\left( {{\hat{\pi}}^{*} \cdot P_{vc}} \right)\delta_{l,{({m^{\prime} - m})}}\delta_{\xi_{v}\xi_{c}}{\int{r\;{{drf}(r)}{R_{v\;\alpha^{\prime}}^{*}(r)}{R_{c\;\alpha}(r)}}}}$where f(r) is the radial component for ϵ_(lp)∫e ^(−ilϕ) e ^(−i(m′-m)ϕ) dϕ=2πδ_(l,(m′-m))∫ξ′ξd ³ r=d _(ξ′ξ)The transition rate between 2-states:

$W_{if} = {\frac{2\pi}{\hslash}{M_{if}}^{2}{\rho(E)}}$Second Quantization for Many Body Dynamics

Second quantization enables multi-body dynamics. Referring now to FIG.76, systems consisting of many identical particles form most of thephysical world. Typical examples are electrons 7602 in a lattice of ions7604 in a solid or nucleons in a heavy nucleus. While the classicaltreatment of identical particles does not differ from the treatment ofnon-identical ones classical theory relies on the assumption that themotion of each individual particle can always be followed.Indistinguishability imposes additional requirements on the quantumtheory of identical particles: vectors representing states must havedefinite symmetry properties with respect to interchanging labels ofidentical particles. Imposing this requirement in the ordinary approachbased on the multiparticle wave function is complicated. Here aconvenient formulation called “second quantization” composed of manyidentical particles, allowing automatic consideration of these symmetryrequirements, is presented. Its most characteristic feature is the useof the creation and annihilation operators in terms of which anyoperator can be expressed and whose action on the system's states isparticularly simple. The true essence of this formulation is theintroduction of the “big” Hilbert space whose vectors can representstates of an arbitrary, also infinite, and even indefinite numbers ofparticles. It is this Hilbert space in which the action of the creationand annihilation operators is naturally defined. Quantum mechanicsformulated using this formalism, when restricted to a subspace of Hcorresponding to a fixed number N of particles (this is possible if thesystem's Hamiltonian commutes with the particle number operator), isfully equivalent to the quantum mechanics based on the N-particleSchrodinger equation supplemented with the appropriate symmetryrequirements. However, the second quantization also opens essentiallynew possibilities. Second quantization constitutes therefore a linkbetween the ordinary quantum mechanics of many-particle systems and therelativistic quantum field theory.

The first quantization framework considers observables as operators andstate as a wave function. In second quantization, states must beconsidered as operators as well. The complete electronic Hamiltonian insecond quantization for the general multi-band case is:

$\mathcal{H} = {{\sum\limits_{b\;\alpha}{\epsilon_{b\alpha}{\mathbb{a}}_{b\alpha}^{\dagger}{\mathbb{a}}_{b\;\alpha}}} + {\sum\limits_{{b\;\alpha},{b^{\prime}\alpha^{\prime}}}{\left\langle {b^{\prime}\alpha^{\prime}{H_{int}}b\;\alpha} \right\rangle{\mathbb{a}}_{b^{\prime}\alpha^{\prime}}^{\dagger}{\mathbb{a}}_{b\;\alpha}}}}$where b, b′ are energy bandsα, α′ are collective index of quantum numbers

^(†) creation operator

${\mathbb{a}}^{\dagger} = {\frac{1}{\sqrt{2}}\left( {\hat{x} + {i\;\hat{p}}} \right)}$

Annihilation operator

${\mathbb{a}} = {\frac{1}{\sqrt{2}}\left( {\hat{x} - {i\;\hat{p}}} \right)}$

$\hat{x} = {\sqrt{\frac{m\;\omega}{\hslash}}x}$$\hat{P} = {{\frac{1}{\sqrt{m\;\omega\;\hslash}}{P\left\lbrack {\hat{x},\hat{P}} \right\rbrack}} = i}$$\hat{H} = {{\frac{1}{\hslash\;\omega}H} = {\frac{1}{2}\left( {{\hat{x}}^{2} + {\hat{P}}^{2}} \right)}}$

Hermitian operators γ₁ and γ₂ may also be represented as:

$\gamma_{1} = {\frac{1}{2}\left( {{\mathbb{a}} + {\mathbb{a}}^{\dagger}} \right)}$$\gamma_{2} = {\frac{1}{2}\left( {{\mathbb{a}} + {\mathbb{a}}^{\dagger}} \right)}$Majorana Photons

Photons in structured light beams (i.e. photons with OAM (orbitalangular momentum)+SAM (spin angular momentum)) and with a finitetransverse size are observed to travel at a speed apparently slower thanthat of light (velocity of light=c), including in vacuum. Photonspropagating in vacuum, however, must still propagate at the speed oflight. This behavior can be explained as a projection effect of theeffective motion of the photon on the axis of propagation of thediverging beam and depends on the geometrical properties of the beam. Inthis scenario, the structured beams carrying OAM, and more specifically,hypergeometric beams, present a group velocity (Vg) that obeys arelationship similar to that proposed by Majorana between spin and massfor bosonic and fermionic relativistic particles that instead involvesOAM, speed of propagation and a virtual mass parameter thatcharacterizes the beam.

Majorana formulated an alternative solution valid for bosonic andfermionic relativistic particles with null or positive-definite restmass in the attempt to avoid the problem of the negative squared masssolution emerging from the Dirac equation that instead were due toanti-electrons, experimentally discovered by Anderson. In ageneralization of the Dirac equation to spin values different than thatof the electron, Majorana found a solution with a denumerable infinitespectrum of particles that obey either the Bose-Einstein or theFermi-Dirac statistics with a precise relationship between spin andmass. The infinite spin solution to the Dirac equation proposed byMajorana gives a spectrum of particles with a positive-definite or nullfinite squared mass. The spectrum of particles described by the Majoranasolution does not have any correspondence with that of the StandardModel. In this scenario, particles and their corresponding antiparticlesare not mutually distinguishable. Since any particle and antiparticlemust have opposite electric charges, this type of solution is valid onlyfor known neutral particles. This relationship applies only to bosonssuch as gravitons and photons, that are zero rest mass particles invacuum and to a few possible exceptions for a class of fermions, theso-called Majorana neutrinos. Clearly, elementary particles such aselectrons and positrons cannot be Majorana particles.

As known from Majorana's works, he progressively came to the idea of therelativistic theory of particles with arbitrary angular momentum. Hestudied first the finite cases for composite systems, and in particulara Dirac-like equation for photons, where he faced the problem bystarting directly from the Maxwell classical field, using the3-dimensional complex vector F=E+iH, where E and H are the electric andmagnetic fields respectively. This expression sets two invariants forthe electromagnetic tensor, one for the real and one for the imaginarypart of F, and introduces a wavefunction of the photon Ψ=E+iH, whoseprobabilistic interpretation is very simple and immediate and reliesdirectly the initial intuition of Einstein-Born: the electromagneticenergy density is proportional to the probability density of thephotons. Maxwell's equations can be written in terms of thiswavefunction; the first is the typical transverse state in quantummechanics for the spin-1 particles, while developing the second Majoranais capable of writing the photon wave equation as a particular case of aDirac equation, which is shown to be equivalent to quantumelectrodynamics (QED). Majorana's theory is ideal for studying boundsystems or photon systems in a structured beam (i.e. photons withOAM+SAM called vector beams).

Examples of Majorana-like “particles” can be found in differentscenarios like in condensed matter physics where composite states ofparticles behave like Majorana fermions, for which a particle and itsantiparticle must coincide and acquire mass from a self-interactionmechanism. This is different from the Standard Model which obeys theHiggs mechanism. These quasiparticles are the product of electromagneticinteractions between electrons and atomic structures present in acondensed matter scenario.

Quasi-particle excitations behaving like Majorana particles have beenobserved in topological superconductors characterized by carrying nullelectric charge and energy. Other types of quasiparticles, calledMajorana zero modes, were observed in Josephson junctions and in solidstate systems. They have potential future applications in informationprocessing and photonic systems.

In addition, structured light beams and other photonic systems canbehave like Majorana particles. They involve not only the spin angularmomentum but also the total angular momentum. Photons, in fact, carryenergy, momentum and angular momentum J. The conserved angular momentumquantity carried by a photon is given by the (vector) sum of the spinangular momentum (SAM), and the orbital angular momentum (OAM).

An example is photons carrying OAM propagating in a structured plasma.They acquire an effective Proca mass through the Anderson-Higgsmechanism and obey a mass/OAM relationship that resembles themathematical structure of the Majorana. The difference between theoriginal solution found by Majorana, based on the space-time symmetriesof the Lorentz group applied to the Dirac equation in vacuum, and thatof photons in a structured plasma, is the Anderson-Higgs mechanism andthe role played by OAM for the mass/total angular momentum relationshipfound for structured electromagnetic beams.

Like what occurs in Majorana's original spin/mass relationship, OAM actsas a term that reduces the total Proca photon mass in the plasma. Whatinduces a Proca mass in photons in a plasma is the breaking of spatialhomogeneity and a characteristic scale length introduced by the plasmastructure at the frequencies at which the plasma is resonant. Thepresence of a characteristic scale length and structures in the plasmapresents strong analogies with the models of space-time characterized bya modified action of the Lorentz group, demonstrating deep analogieswith the dynamics described by the Dirac equation when a characteristicscale length is present. Fermionic and anionic-like behaviors withnon-abelian gauge structure are obtained through the coherentsuperposition of OAM states. There are possible formulations withnon-abelian gauge theories.

OAM and Majorana Photons

The slowing down of a structured light beam in vacuum is related togeometrical divergence of the beam. This is a different mechanism inwhich light being slowed due to the presence of matter. Structured beamphotons propagating in a vacuum may show a different behavior than thatof a plane wave because of the field confinement induced by the finiteextent and the structure of the beam that changes the wave vector withthe result of altering the group velocity, V_(g), measured along thepropagation axis.

Phase velocity V_(p) is also different for different OAM modes. The coreidea regarding the speed of propagation of OAM Eigen modes is that K_(z)is not the only component of the wave vector that must be considered inorder to have a complete description of the system.

In cylindrical coordinates of OAM E(ρ,ϕ, z), where ρ=radial,ϕ=azimuthal, z=vertical

$k = \sqrt{k_{\rho}^{2} + k_{\phi}^{2} + k_{z}^{2}}$$k_{\phi} = \frac{l}{\rho}$E(ρ,ϕ,z,t)=R(ρ)Φ(ϕ)Z(z)T(t)E ₀ in general

${\Phi(\phi)} = {\sum\limits_{l = {- \infty}}^{\infty}{c_{l}e^{{jl}\;\phi}}}$

At any specific radial mode:

${\overset{\rightarrow}{k}\left( \rho_{0} \right)} = {{k_{\rho}\hat{\rho}} + {\frac{l}{\rho_{0}}\hat{\phi}} + {k_{z}\hat{z}}}$and phase velocity:

$v_{p} = {\frac{w}{k} = \frac{w}{\sqrt{k_{\rho}^{2} + k_{\phi}^{2} + k_{z}^{2}}}}$but we also know that v_(g)v_(p)=c², where v_(p)=phase velocity andv_(g)=group velocity

For OAM beams:

$v_{p} = \frac{c}{\left( {1 - \frac{k_{\rho}^{2}}{2k_{0}^{2}}} \right)}$$v_{g} = {c\left( {1 - \frac{k_{\rho}^{2}}{2k_{0}^{2}}} \right)}$

It is possible to calculate the exact group velocity of a paraxial beamat any point in space using the wave picture description. In the wavedescription, the group velocity is given by vg=|∂ω∇Φ|⁻¹, where Φ(r)represents the wave phase front. For Laguerre-Gauss modes, the groupvelocity along z, which has an explicit dependence on the propagationdistance z, is given by:

${v_{g}(z)} = \frac{c}{1 + {\left( \frac{\theta_{0}}{4} \right)^{2}\left( {{2p} + {l} + 1} \right){F\left( \frac{z}{z_{R}} \right)}}}$${F\left( \frac{z}{z_{\;^{R}}} \right)} = \frac{1 + {6\left( \frac{z}{z_{R}} \right)^{2}} - {3\left( \frac{z}{z_{R}} \right)^{4}}}{\left\lbrack {1 + \left( \frac{z}{z_{R}} \right)^{2}} \right\rbrack^{3}}$where

$z_{R} = \frac{8}{k\theta_{0}^{2}}$(Rayleigh distance).

This relationship reflects the strange geometry of these beams. The timedelays or, better, different group velocities can be interpreted as theeffect of a fictitious mass m_(v) of a quasiparticle state thatcharacterizes the dynamics of the beam and is described by aSchrodinger-like equation at low velocities propagating with the groupvelocity V_(g):

$v_{g} = {{\frac{1}{\hslash}{\nabla_{k_{z}}\frac{\hslash^{2}k_{z}^{2}}{2m_{v}}}} = {\frac{\hslash\; k_{z}}{m_{v}} = \frac{p}{m_{v}}}}$Thus:

$m_{v} = {\frac{\hslash k_{z}}{c}\left\lbrack {1 + {\left( \frac{\theta_{0}}{1} \right)^{2}\left( {{2p} + {l} + 1} \right){F\left( \frac{z}{z_{R}} \right)}}} \right\rbrack}$

This fictitious mass term leads to Majorana mass M. Any of thesequasiparticle states have a precise relationship between their angularmomentum value and their fictitious mass in vacuum that resembles thebehavior of Proca photons with OAM in a plasma.

This means that the process of beam confinement and structuring plays arole similar to that of the Anderson-Higgs mechanism that induces theProca mass on the photons together with their angular momentum/massrelationship in a Majorana model, similar to what occurs in waveguideseven if the beams are freely propagating in space.

Therefore, photons propagating in vacuum belonging to this class ofstructured beams obey the mathematical rules of Majorana model ofquasiparticle states. In fact, by comparing the two group velocities oneobtains a relationship that involves the wavevector K and the fictitiousmass term m_(v)

$M = {\frac{\hslash k_{Z}}{c} = \frac{m_{v}}{\left\lbrack {1 + {\left( \frac{\theta_{0}}{4} \right)^{2}\left( {{2p} + {l} + 1} \right){F\left( \frac{z}{z_{R}} \right)}}} \right\rbrack}}$The corresponding Energy is:W ₀ =Mc ² =cℏk _(z)

Each of these beams behaves as a particle in the low energy limit thatobey a Schrodinger/Dirac equation with a fictitious Majorana mass M anda Majorana-model OAM/fictitious mass relationship as in the 1932 paperbyMajorana. The M in Majorana's original work was

$M = \frac{2m_{v}}{s^{*} + \frac{1}{2}}$where

$s^{*} = {{\theta_{0}^{2}\left( {{2p} + {l} + 1} \right)}{F\left( \frac{z}{z_{R}} \right)}}$is the angular momentum part of the virtual mass.

Therefore, the dynamics and structure of the beam is uniquelycharacterized in space and time through the rules of the Poincare groupthat build up the Majorana model. Particles with opposite OAM valuesdiffer by chirality even if they are propagating with the same groupvelocity. Through the projection effect along the axis of propagation,one can resize and shape the beam at will to obtain different apparentsub-luminal velocities for different values of OAM and spatially bufferinformation in time.

Braiding of Majorana Fermions with Majorana Photons (Vector Vortex Beamswith OAM)

Braiding of non-Abelian anyons is what is needed in topological quantuminformatics and quantum computing to improve entanglement properties ofquantum computing components. One promising class of non-abelian anyonsare the Majorana bound states (MBS) that emerge in topologicalsuperconductors as zero-energy quasiparticles. In recent years therehave been many demonstrations of detecting and manipulating MBS incondensed matter platforms. Implementations based on one-dimensional(1D) semiconducting wires (SW) are the most seen demonstration. Severalexperiments have reported characteristic transport signatures in theform of a zero-bias conductance peak compatible with the presence ofzero-energy MBS.

The most important feature of MBS is their exchange, or braidingstatistics: moving these Majorana quasiparticles around each other andexchanging their positions will implement non-Abelian unitarytransformations that depend only on the topology of the trajectories.This way, information can be encoded in the twist of the state function(wave function). This is generally illustrated in FIG. 77. A light beam7702 is provided to an OAM generator to have OAM values applied to thelight beam. The OAM infused light beam is then applied to a braidingprocess 7706 to braid the anyons of the solid state material 7708 toencode the information in the twists of the state functions. Suchunitary transforms are more robust against decoherence and dephasing dueto local environments, as opposed to quantum computing with conventionalqubits, which would make this approach interesting for topologicalquantum computing. In order to detect the Majorana signatures, as wellas to manipulate the braiding of the MBS, one must look for interferenceschemes (lift the ground state degeneracy), making MBS interact.

It is hypothesised that there is a Clifford algebra generalization ofthe quaternions and its relationship with braid group representationsrelated to Majorana fermions. That is the topological quantum computingis based on the fusion rules for a Majorana fermion. In this constructMajorana fermions can be seen not only in the structure of co-ops ofelectrons (i.e. quantum Hall effect), but also in the structure ofsingle electrons by the decomposition of the operator algebra for afermion into a Clifford algebra generated by two Majorana operators.These braiding representations have important applications in quantuminformatics and topology. Majorana operators give rise to a particularlyrobust representation of the braid group that is then furtherrepresented to find the phases of the fermions under their exchanges ina plane space. The braid group (represented by the Majorana operators)represents exchanges of Majoranas in space.

Examples of braiding of anyons 7802 are illustrated in FIG. 78. Here themathematics will form a bridge between theoretical models of anyons andtheir applications to quantum computing components. The braiding ofMajorana fermions are natural representations of Clifford algebras andalso with the representations of the quaternions as SU (2) to the braidgroup. This could also explain the vortices of the quantum Hall effect.It may not work for electrons in one-dimensional nanowires. However,there are three Majorana operators, generating a copy of thequaternions.

Referring not to FIG. 79, a braid 7902 is an embedding of a collectionof strands 7904 that have their ends 7906 in two rows of points that areset one above the other with respect to a choice of vertical. Thestrands 7904 are not individually knotted, and they are disjoint fromone another. Braids 7902 can be multiplied by attaching the bottom row7906B of one braid to the top row 7904A of the other braid as shown inFIG. 80. This multiplication of braids 7902 form a group under thisnotion of multiplication. Thus, the theory of braids is critical to thetheory of knots and links for both electromagnetic knots as well asquantum wave function or state knots.

Braiding with a Y-Junctions with Vector Beams carrying Orbital AngularMomentum

It is possible to perform braiding of Majorana fermions with Majoranaphotons which are the vortex vector beams that carry orbital angularmomentum. In this scenario, there would be four Majoranas (M₀, M₁, M₂,M₃) where M₁, M₂, M₃ are considered to be at the ends of the Y-junctionand M₀ at the center of the Y-junction (Y-junction waveguide).Sophisticated braiding may be performed by placing the superconductingY-Junctions in a cavity with an OAM beam acting on the Y-junction. Therewill be a Berry's phase contribution to this dynamic in this setup. TheHamiltonian can then be constructed with a rotating wave approximation.The Hamiltonian for this case couples the states with photons in thecavity at the cavity's frequency.

Quaternion Implementation of Wave Function Knots and ElectromagneticKnots

Fermions

For Fermions:Ψ²=(Ψ^(†))²=0 and ΨΨ^(†)+Ψ^(†)Ψ=1If there are two Majorana Fermions, then:xy=−yx

Majoranas are their own anti-particle and thereforex=x ^(†) x ²=1xy+yx=0 y ²=1

Therefore one can make one Fermion from two Majoranas ifΨ=½=(x+iy)Ψ^(†)=½(x−iy)Then:

$\begin{matrix}{\Psi^{2} = {\Psi\Psi}^{\dagger}} \\{= {\frac{1}{4}\left( {x + {iy}} \right)\left( {x - {iy}} \right)}} \\{= {{x^{2} - y^{2} + {i\left( {{xy} + {yx}} \right)}} = 0}}\end{matrix}$

Also, one can make two Majoranas from one Fermion according to:

$x = {\frac{1}{2}\left( {\Psi + \Psi^{\dagger}} \right)}$$y = {\frac{1}{2i}\left( {\Psi - \Psi^{\dagger}} \right)}$

Now with respect to three Majoranas, three Majoranas can represent aquaternion group. If there are Majoranas x, y, z:

Then x²=1, y²=1, z²=1

Let I=yx J=zy K=xz

Then:

I²−1 J²=−1 K²=−1 IJK=−1

The operators may then be defined:

$A = {\frac{1}{\sqrt{2}}\left( {1 + I} \right)}$$B = {\frac{1}{\sqrt{2}}\left( {1 + J} \right)}$$C = {\frac{1}{\sqrt{2}}\left( {1 + K} \right)}$

Can braid one another as

ABA=BAB BCB=CBC ACA=CAC

These braiding operators are entangling and can be used for universalquantum computation.

${ABA} = {{BAB} = {\frac{1}{\sqrt{2}}\left( {I + J} \right)}}$The Clifford algebra for Majoranas fixes the representation of thebraiding. The braiding of Majoranas is not restricted to a nano-wire andin principle could be applied to 2-strands, 3-strands, 4-strands orn-strands. The 3-strands will represent a quaternion case.

In a 2-qubit state:|ϕ

==a|00

+b|01

+c|10

+d|11

is entangled when ad−bc≠0.SU(2) Framework for Braiding

A matrix SU(2) has a form:

$M = \begin{bmatrix}z & w \\{- w^{*}} & z^{*}\end{bmatrix}$

To be in SU(2) Det(M)=1 and M_(†)=M⁻¹

If z=a+ib and w=c+id i=√{square root over (−1)}

Then

$M = \begin{bmatrix}{a + {ib}} & {c + {id}} \\{{- c} + {id}} & {a - {ib}}\end{bmatrix}$With a²+b²+c²+d²=1Then:

$M = {{a\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}} + {b\begin{bmatrix}i & 0 \\0 & {- i}\end{bmatrix}} + {c\begin{bmatrix}0 & 1 \\{- 1} & 0\end{bmatrix}} + {d\begin{bmatrix}0 & i \\i & 0\end{bmatrix}}}$Where

${1 = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}};{I = \begin{bmatrix}i & 0 \\0 & {- i}\end{bmatrix}};{J = \begin{bmatrix}0 & 1 \\{- 1} & 0\end{bmatrix}};{{{and}\mspace{20mu} K} = \begin{bmatrix}0 & i \\i & 0\end{bmatrix}}$And

I² = −1 J² = −1 K² = −1 IJK = −1 IJ = K JK = I KI = J JI = −K KJ = −I IK= −J

The algebra of 1,I, J, K is a quaternion algebra. In general:q=a+bI+cJ+dKq ^(†) =a−bI−cJ−dKqq ^(†) =a ² +b ² +c ² +d ²=length of q=√{square root over (qq ^(†))} unit q=√{square root over (qq^(†))}=1

If U & V are pure quaternions, (a=0), then:uv=−u·v+u×v

Quaternions are intimately connected to topology and braiding isfundamental to Majoranas and their application to quantum computing. Thesame quatemionic algebra can also be used to create electromagneticknots.

Referring now to Table 1 and Table 2 herein below there areelectromagnetic equations in traditional U(1) symmetry as well as SU(2)symmetry. With temperature control, one can go from one symmetry spaceto another and the quaternionic algebra used for braiding wave functionscan be used to describe these states as well.

TABLE 1 U(1) Symmetry Form(Traditional Maxwell Equations) SU(2) SymmetryForm Gauss′ ∇ · E = J₀ ∇ · E = J₀ − iq(A · E − E · A) Law Ampère's Law${\frac{\partial E}{\partial t} - {\nabla{\times B}} - J} = 0$${\frac{\partial E}{\partial t} - {\nabla{\times B}} - J + {{jq}\left\lbrack {A_{0},E} \right\rbrack} - {{iq}\left( {{A \times B} - {B \times A}} \right)}} = 0$∇ · B = 0 ∇ · B + iq(A · B − B · A) = 0 Faraday's Law${{\nabla{\times E}} + \frac{\partial B}{\partial t}} = 0$${{\nabla{\times E}} + \frac{\partial B}{\partial t} + {{iq}\left\lbrack {A_{0},B} \right\rbrack}} = {{{iq}\left( {{A \times E} - {E \times A}} \right)} = 0}$

TABLE 2 U(1) Symmetry Form (Traditional Maxwell Theory) SU(2) SymmetryForm ρ_(e) = J₀ ρ_(e) = J₀ − iq(A · E − E · A) = J₀ + qJ_(z) ρ_(m) = 0ρ_(m) = −iq(A · B − B · A) = −iqJ_(y) g_(e) = J g_(e) = iq[A₀, E] − iq(A× B − B × A) + J = iq[A₀, E] − iqJ_(x) + J g_(m) = 0 g_(m) = iq[A₀, B] −iq(A × E − E × A) = iq[A₀, B] − iqJ_(z) σ = J/E$\sigma = {\frac{\left\{ {{{iq}\left\lbrack {A_{0},E} \right\rbrack} - {{iq}\left( {{A \times B} - {B \times A}} \right)} + J} \right\}}{E} = \frac{\left\{ {{{iq}\left\lbrack {A_{0},E} \right\rbrack} - {iqJ}_{x} + J} \right\}}{E}}$s = 0$s = {\frac{\left\{ {{{iq}\left\lbrack {A_{0},B} \right\rbrack} - {{iq}\left( {{A \times E} - {E \times A}} \right)}} \right\}}{H} = \frac{\left\{ {{{iq}\left\lbrack {A_{0},B} \right\rbrack} - {iqJ}_{z}} \right\}}{H}}$New OAM and Matter Interactions with Graphene Honeycomb Lattice

Due to crystalline structure, graphene behaves like asemi-metallicmaterial, and its low-energy excitations behave as massless Diracfermions. Because of this, graphene shows unusual transport properties,like an anomalous quantum haleffect and Klein tunneling. Its opticalproperties are strange: despite being one-atom thick, graphene absorbs asignificant amount of white light, and its transparency is governed bythe fine structure constant, usually associated with quantumelectrodynamics rather than condensed matter physics.

Current efforts in study of structured light are directed, on the onehand, to the understanding and generation of twisted light beams, and,on the other hand, to the study of interaction with particles, atoms andmolecules, and Bose-Einstein condensates.

Thus, as shown in FIG. 81, an OAM light beam 8102 may be interacted withgraphene 8104 to enable the OAM processed photons to alter the state ofthe particles in the graphene 8104. The interaction of graphene 8104with light 8102 has been studied theoretically with differentapproaches, for instance by the calculation of optical conductivity, orcontrol of photocurrents. The study of the interaction of graphene 8104with light carrying OAM 8102 is interesting because the world is movingtowards using properties of graphene for many diverse applications.Since the twisted light 8102 has orbital angular momentum, one mayexpect a transfer of OAM from the photons to the electrons in graphene8104. However, the analysis is complicated by the fact that thelow-lying excitations of graphene 8104 are Dirac fermions, whose OAM isnot well-defined. Nevertheless, there is another angular momentum, knownas pseudospin, associated with the honeycomb lattice of graphene 8104,and the total angular momentum (orbital plus pseudospin) is conserved.

The interaction Hamiltonian between OAM and matter described herein canbe used to study the interaction of graphene with twisted light andcalculate relevant physical observables such as the photo-inducedelectric currents and the transfer of angular momentum from light toelectron particles of a material.

The low-energy states of graphene are two-component spinors. Thesespinors are not the spin states of the electron, but they are related tothe physical lattice. Each component is associated with the relativeamplitude of the Bloch function in each sub-lattice of the honeycomblattice. They have a SU(2) algebra. This degree of freedom ispseudospin. It plays a role in the Hamiltonian like the one played bythe regular spin in the Dirac Hamiltonian. It has the same SU(2) algebrabut, unlike the isospin symmetry that connects protons and neutrons,pseudospin is an angular momentum. This pseudospin would be pointing upin z (outside the plane containing the graphene disk) in a state whereall the electrons would be found in A site, while it would be pointingdown in z if the electrons were located in the B sub-lattice.

Interaction Hamiltonian OAM with Graphene Lattice (Honeycomb)

Referring now to FIG. 82, there is illustrated a Graphene lattice in aHoneycomb structure. If T₁ and T₂ are the primitive vectors of theBravais lattice and k and k′ are the corners of the first Brillouinzone, then the Hamiltonian is:

${H_{0}(k)} = {t\begin{bmatrix}0 & {1 + e^{{- {ik}} \cdot T_{2}} + e^{{- {ik}} \cdot {({T_{2} - T_{1}})}}} \\{1 + e^{{ik} \cdot T_{2}} + e^{{ik} \cdot {({T_{2} - T_{1}})}}} & 0\end{bmatrix}}$The carbon atom separation on the lattice is a=1.42 Å. If this matrix isdiagonalized, the energy Eigen values representing energy bands ofgraphene is obtained.

${E_{\pm}(k)} = {{\pm t}\sqrt{2 + {2\mspace{11mu}{\cos\left( {\sqrt{3}k_{y}a} \right)}} + {4\;{\cos\left( {\frac{\sqrt{3}}{2}k_{y}a} \right)}{\cos\left( {\frac{3}{2}k_{x}\alpha} \right)}}}}$If k=K+(q_(x), q_(y))Then

${H_{0}^{K}(q)} = {{\frac{3{ta}}{2}\begin{pmatrix}0 & {q_{x} + {iq}_{y}} \\{q_{x} - {iq}_{y}} & 0\end{pmatrix}\mspace{14mu}{for}\mspace{14mu} q_{x}a} ⪡ {1\mspace{20mu} q_{y}a} ⪡ 1}$

So for 2D-Hamiltonian:H ₀ ^(α)(q)=

v _(f)α(Υ_(x) q _(x)−σ_(y) q _(y))σ=(σ_(x),σ_(y)) Pauli matricesα=±1

Where Fermi velocity equals:

$v_{f} = {\frac{3{at}}{2\hslash} \sim {300\mspace{14mu}{times}\mspace{14mu}{slower}\mspace{14mu}{than}\mspace{14mu} c}}$The Eigen states of these Hamiltonians are spinors with 2-componentswhich are 2 elements of lattice base.

For circular graphene of radius r₀, the low-energy states can be foundin cylindrical coordinates with

${\Psi_{mv}\left( {r,\theta} \right)} = {\frac{N_{mv}}{2\pi}{J_{m}\left( {q_{mv}r} \right)}e^{{im}\;\theta}}$$q_{mv} = \frac{x_{mv}}{r_{0}}$where x_(mv) has a zero of J_(m)(X)

To study the interaction of graphene with OAM beam, the z-component ofOAM operator

$L_{z} = {{- i}\;\hslash\frac{d}{d\;\theta}1}$is examined and then the commutation relationship between Hamiltonianand OAM is:[H ₀ ^(α) ,L _(z)]=−i

v _(f)α(σ_(x) P _(x)+σ_(y) P _(y))

To construct a conserved angular momentum, pseudospin is added to L_(z)and the total angular momentum is:

$J_{z}^{\alpha} = {L_{z} - {\alpha\frac{\hslash}{2}\sigma_{z}}}$

This operator does commute with Hamiltonian and

${J_{z}\Psi_{{mv},k}} = {\left( {m + \frac{1}{2}} \right)\hslash\;\Psi_{{mv},k}}$near  k  or  k^(′)Interaction Hamiltonian

We know the vector potential for OAM beam in Coulomb gauge is:

${A\left( {r,t} \right)} = {{A_{0}{e^{i{({{q_{z}z} - {\omega\; t}})}}\left\lbrack {{\epsilon_{\sigma}{J_{l}({qr})}e^{{il}\;\theta}} - {\sigma\; i\;\hat{z}\frac{q}{q_{z}}{J_{l + \sigma}\left( q_{r} \right)}e^{{i{({l + \sigma})}}\theta}}} \right\rbrack}} + \ldots}$where ∈_(σ)={circumflex over (x)}+iσŷ polarization vectors σ=±1The radial part of the beam are Bessel functions J_(l)(gr) andJ_(l+r)(qr). A Laguerre-Gaussian function could also be used instead ofBessel functions. Here q_(z) and q are for structured light and q_(x),q_(y), q_(mv) are for electrons.

To construct the interaction Hamiltonian:{right arrow over (P)}→{right arrow over (P)}+e{right arrow over (A)}Then

$\begin{matrix}{H^{\alpha} = {{\hslash\; v_{f}{\alpha\left( {{\sigma_{x}q_{x}} - {\sigma_{y}q_{y}}} \right)}} + {{ev}_{f}{\alpha\left( {{\sigma_{x}A_{x}} - {\sigma_{y}A_{y}}} \right)}}}} \\{= {H_{0}^{\alpha} + H_{int}^{\alpha}}}\end{matrix}$Since Graphene is 2D, there are only x, y values of the EM field.

At z=0 (Graphene disk), the vector potential is:A(r,θ,t)=A ₀({circumflex over (x)}+iσŷ)e ^(−iωt) J _(l)(qr)e ^(ilθ)+ . ..

Let's have A₊=A₀e^(−iωt)J_(l)(qr)e^(ilθ) Absorption of 1 photon

A⁻=A₀e^(−iωt)J_(l)(qr)e^(−ilθ) emission of 1 photon

Then the interaction Hamiltonian close to a Dirac point a is:

$H_{int}^{\alpha,\sigma} = {{ev}_{f}\begin{bmatrix}0 & {{\left( {\sigma - \alpha} \right)A_{+}} + {\left( {\alpha + \sigma} \right)A_{-}}} \\{{\left( {\alpha + \sigma} \right)A_{+}} + {\left( {\alpha - \sigma} \right)A_{-}}} & 0\end{bmatrix}}$For α=1 (near K) and σ=+1

$H_{int}^{K +} = {2{{ev}_{f}\begin{bmatrix}0 & A_{-} \\A_{+} & 0\end{bmatrix}}}$For α=−1 (near K′) and σ=+1

$H_{int}^{K^{\prime} +} = {2{{ev}_{f}\begin{bmatrix}0 & A_{+} \\A_{-} & 0\end{bmatrix}}}$Then the transition matrix becomes:

$\begin{matrix}{M_{if} = \left\langle {i{H_{int}}f} \right\rangle} \\{= \left\langle {c,m^{\prime},v^{\prime},{\alpha{H_{int}^{\alpha\;\sigma}}v},m,v,\alpha} \right\rangle} \\{= {\int{{\Psi_{m^{\prime}v^{\prime}\alpha}^{c\;\dagger}\left( {r,\theta} \right)}{H_{int}^{\alpha\;\sigma}\left( {r,\theta} \right)}{\Psi_{{mv}\;\alpha}^{v}\left( {r,\theta} \right)}r\; d\; r\; d\;\theta}}}\end{matrix}$Where near K:

$\Psi_{mvK}^{v} = {\frac{1}{\sqrt{2}}\begin{pmatrix}\Psi_{{m + 1},v} \\{i\;\Psi_{mv}}\end{pmatrix}}$ $\Psi_{mvK}^{v} = {\frac{1}{\sqrt{2}}\begin{pmatrix}\Psi_{{m + 1},v} \\{{- i}\;\Psi_{mv}}\end{pmatrix}}$Where

$\Psi_{mv} = {\left( {r,\theta} \right) = {\frac{N_{mv}}{2\pi}{J_{m}\left( {q_{mv}r} \right)}e^{{im}\;\theta}}}$as before

$W_{if} = {\frac{2\pi}{\hslash}{M_{if}}^{2}{\rho(E)}\mspace{14mu}{transition}\mspace{14mu}{rate}}$OAM-Assisted Crisper (A New OAM and Matter Interactions) for GeneEditing and Molecular Manipulations

The most advanced CRISPR technology today is CRISPR-Cas9. CRISPR-Cas9 isa technology using an enzyme that uses CRISPR sequences as a guide torecognize and cleave specific strands of DNA that are complementary tothe CRISPR sequence. It was created from a naturally occurring processof genome editing system in bacteria. The bacteria capture sections ofDNA from attacking viruses and use them to create DNA segments (CRISPRarrays). The bacteria produce RNA segments from the CRISPR arrays totarget the viruses' DNA. The bacteria then use Cas9 to cut the DNAapart, which disables the virus. Cas9 also works in the lab. Scientistscreate a small piece of RNA with a short “guide sequence” that attaches(binds) to a specific target sequence of DNA in a genome. The labcreation of this engineered guide sequence is where OAM can come intoplay. As in bacteria, the modified RNA is used to recognize the DNAsequence, and engineered sequence cuts the DNA at the targeted location.Once the DNA is cut, the cell's own DNA repair machinery can be used toadd or delete pieces of genetic material, or to make changes to the DNAby replacing an existing segment with a customized DNA sequence. Thus,OAM-Assisted CRISPR can be used to modify RNA or cut the DNA at thetargeted location.

Currently, most research on genome editing is done to understanddiseases using cells and animal models. It is being explored in researchon a wide variety of diseases, including single-gene disorders such ascystic fibrosis, hemophilia, and sickle cell disease. It also can beused for the treatment and prevention of more complex diseases, such ascancer, heart disease, mental illness, and HIV infection.

Most of the changes introduced with gene editing are limited to somaticcells, which are cells other than egg and sperm cells. These changesaffect only certain tissues and are not passed from one generation tothe next. However, changes made to genes in egg or sperm cells (germlinecells) or in the genes of an embryo could be passed to futuregenerations.

OAM can be used in the lab to produce the “guide sequence” for geneediting called OAM-Assisted CRISPR. The non-bacterial method of usingOAM in the lab to produce the “guide sequence” is well within reach withan OAM-Assisted CRISPR. OAM-Assisted CRISPR can impact both precisiongenetics as well as precision pharma industries.

Thus as shown in FIG. 83, a combination of a CRISPR-Cas9 system 8302 anda OAM infused light beam 8304 may be applied to a gene sequence 8306.The OAM assisted CRISPR-Cas9 system would cause removal a desired genesequence 8308 and provide a revised gene sequence 8310 with the removedsequence 8308 that may be replaced by another sequence in a desiredmanner.

It will be appreciated by those skilled in the art having the benefit ofthis disclosure that this quantum mechanical framework for interactionof OAM with applications in solid states, biosciences and quantumcomputing provides an improved manner for enabling the use of OAMprocessed light beams for imparting OAM to electrons of various types ofmaterials. It should be understood that the drawings and detaileddescription herein are to be regarded in an illustrative rather than arestrictive manner, and are not intended to be limiting to theparticular forms and examples disclosed. On the contrary, included areany further modifications, changes, rearrangements, substitutions,alternatives, design choices, and embodiments apparent to those ofordinary skill in the art, without departing from the spirit and scopehereof, as defined by the following claims. Thus, it is intended thatthe following claims be interpreted to embrace all such furthermodifications, changes, rearrangements, substitutions, alternatives,design choices, and embodiments.

What is claimed is:
 1. A method for applying orbital angular momentum(OAM) to electrons of a semiconductor material, comprising: generating aplane wave light beam; applying at least one orbital angular momentum tothe plane wave light beam to generate an OAM light beam; controllingtransitions of electrons between quantized states within thesemiconductor material to perform quantum entanglement within thesemiconductor material responsive to the at least one orbital angularmomentum applied to the plane wave light beam; and transmitting the OAMlight beam at the semiconductor material to induce the transitions ofthe electrons between the quantize states and perform the quantumentanglement within the semiconductor material.
 2. The method of claim1, wherein the step of controlling further comprises the step ofcontrolling at least one of spectrum, pulse shape, time of arrival,transverse wave vector and position of quantized states of photonswithin the OAM light beam to control the transition of electrons betweenthe quantized states.
 3. The method of claim 1, wherein the step ofcontrolling further comprises the steps of measuring a state of at leastone photon within the OAM light beam to cause states of remainingphotons to collapse, wherein the collapse of the states of the remainingphotons enable the remaining photons to interact and entangle.
 4. Themethod of claim 1, wherein step of controlling further comprises furthercomprises creating qudits responsive to the at least one orbital angularmomentum applied to the plane wave light beam.
 5. The method of claim 4,wherein the step of controlling further comprises creating a higherorder radix system to create the qudits responsive to the at least oneorbital angular momentum applied to the plane wave light beam.
 6. Themethod of claim 1, wherein the induced transitions of the electronscreate a net electric current and magnetic field.
 7. The method of claim1, wherein the step of controlling further controls the transitions ofelectrons between the quantized states using OAM-generalizedsemiconductor Bloch equations and coherences of the electrons.
 8. Themethod of claim 1, wherein the step of controlling further usescylindrical coordinates to decouple Heisenberg equations.
 9. The methodof claim 1, wherein the OAM light beam is within frequencies from aninfrared spectrum to a UV spectrum.
 10. The method of claim 1, whereinthe step of controlling further comprises spatially bufferinginformation in time by controlling a size and shape of the OAM lightbeam to obtain different apparent sub-liminal velocities for differentvalues of OAM.
 11. The method of claim 1, wherein the step ofcontrolling further comprises encoding information in a twist of a statefunction.
 12. The method of claim 11, wherein the step of encodingfurther comprises the step of exchanging positions of Majoranaquasiparticles in a non-Abelian unitary transformation depending only ona topology of trajectories of the Majorana quasiparticles.
 13. Themethod of claim 1, wherein the step of controlling further comprises thestep of braiding Majorana fermion with Majorana photons that arecarrying the orbital angular momentum within the OAM light beam.
 14. Themethod of claim 1, wherein the step of controlling further comprisesusing temperature control to go from one symmetry space to another tobraid wave functions within the OAM light beam.
 15. A system forapplying orbital angular momentum (OAM) to electrons of a semiconductormaterial, comprising: a light source generator for generating a planewave light beam; orbital angular momentum (OAM) processing circuitry forapplying at least one orbital angular momentum to the plan wave lightbeam to generate an OAM light beam, wherein the OAM processing circuitrycontrols transitions of electrons between quantized states within thesemiconductor material to perform quantum entanglement within thesemiconductor material responsive to the at least one orbital angularmomentum applied to the plane wave light beam; and a transmitter fortransmitting the OAM light beam at the semiconductor material to inducethe transitions of the electrons between the quantize states and performthe quantum entanglement within the semiconductor material.
 16. Thesystem of claim 15, wherein the OAM processing circuitry controls atleast one of spectrum, pulse shape, time of arrival, transverse wavevector and position of quantized states of photons within the OAM lightbeam to control the transition of electrons between the quantizedstates.
 17. The system of claim 15, wherein the OAM processing circuitrymeasures a state of at least one photon within the OAM light beam tocause states of remaining photons to collapse, wherein the collapse ofthe states of remaining photons enable the remaining photons to interactand entangle.
 18. The system of claim 15, wherein OAM processingcircuitry creates qudits responsive to the at least one orbital angularmomentum applied to the plane wave light beam.
 19. The system of claim18, wherein the OAM processing circuitry creates a higher order radixsystem to create the qudits responsive to the at least one orbitalangular momentum applied to the plane wave light beam.
 20. The system ofclaim 15, wherein the induced transitions of the electrons create a netelectric current and magnetic field.
 21. The system of claim 15, whereinthe OAM processing circuitry controls the transitions of electronsbetween quantized states using OAM-generalized semiconductor Blochequations and coherences of the electrons.
 22. The system of claim 15,wherein the OAM processing circuitry uses cylindrical coordinates todecouple Heisenberg equations.
 23. The system of claim 15, wherein theOAM light beam is within frequencies from an infrared spectrum to a UVspectrum.
 24. The system of claim 15, wherein the OAM processingcircuitry spatially buffers information in time by controlling a sizeand shape of the OAM light beam to obtain different apparent sub-liminalvelocities for different values of OAM.
 25. The system of claim 15,wherein the OAM processing circuitry further encodes information in atwist of a state function.
 26. The system of claim 25, wherein the OAMprocessing circuitry further encodes by exchanging positions of Majoranaquasiparticles in a non-Abelian unitary transformation depending only ona topology of trajectories of the Majorana quasiparticles.
 27. Thesystem of claim 15, wherein the OAM processing circuitry braids Majoranafermion with Majorana photons that are carrying the orbital angularmomentum within the OAM light beam.
 28. The system of claim 15, whereinthe OAM processing circuitry uses temperature control to go from onesymmetry space to another to braid wave functions within the OAM lightbeam.